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STUDY PACKAGE
Subject : Mathematics
Topic : DIFFRENTIATION
Available Online : www.MathsBySuhag.com

Index
1. Theory
2. Short Revision
3. Exercise (Ex. 1 + 5 = 6)
4. Assertion & Reason
5. Que. from Compt. Exams
6. 39 Yrs. Que. from IIT-JEE(Advanced)
7. 15 Yrs. Que. from AIEEE (JEE Main)

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Differentiation
A. First Principle Of Differentiation
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1. The derivative of a given function f at a point x = a on its domain is defined as:

Limit f (a h)f (a) , provided the limit exists & is denoted by f (a).
h 0
h
f ( x )f (a)
i.e. f (a) = Limit
x a , provided the limit exists.
x a

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2. If x and x + h belong to the domain of a function f defined by y = f(x), then
Limit f ( x h)f ( x ) if it exists, is called the Derivative of f at x & is denoted by f (x) or dy . i.e., f (x) = Limit
h 0 dx h 0
h
f ( x h)f ( x )
This method of differentiation is also called ab-initio method or first principle.
h
Solved Example # 1 Find derivative of following functions by first principle
(i) f(x) = x 2 (ii) f(x) = tan x (iii) f(x) = esinx
2 2 2
( x h) x lim 2xh h = 2x.
Solution (i) f(x) = hlim 0 = h0
h h

tan( x h) tan x
(ii) f(x) = hlim
0
h
lim tan( x h x )[1 tan x tan( x h)] tan h
= h0 = hlim
0 . (1 + tan2x) = sec2x.
h h
sin ( x h ) sin x
e e
(iii) f(x) = hlim
0
h

= hlim e sin x

e sin ( x h )sin x 1 sin( x h) sin x

0 h
sin( x h) sin x
sin( x h) sin x
= esin x hlim
0 = esin x cos x
h
3. Differentiation of some elementary functions
f(x) f(x)
1. x n nx n 1 (x R, n R)
2. ax ax n a
1
3. n |x|
x
1
4. logax
x n a
5. sin x cos x
6. cos x sin x
7. sec x sec x tan x
8. cosec x cosec x cot x
9. tan x sec2 x
10. cot x cosec x
4. Basic Theorems
d
1. (f g) = f(x) g(x)
dx
d d
2. (k f(x)) = k f(x)
dx dx
d
3. (f(x) . g(x)) = f(x) g(x) + g(x) f(x)
dx
d f ( x ) g( x ) f ( x) f ( x ) g( x )
4. g( x) =
dx g2 ( x )
d
5. (f(g(x))) = f(g(x)) g(x)
dx
dy dy dz
This rule is also called the chain rule of differentiation and can be written as = .
dx dz dx
dy dy dx
Note that an important inference obtained from the chain rule is that =1= .
dy dx dy
dy 1
=
dx
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another way of expressing the same concept is by considering y = f(x) and x = g(y) as inverse functions of each
other.
dy dx 1
= f(x) and = g(y) g(y) =
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dx dy f ( x)
Solved Example # 2
Find the differential of the following functions with respect to x.
x
(i) f(x) = esin x (ii) f(x) = sin( 2x 3) (iii) f(x) = (iv) f(x) = x . sin x
1 x2

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Solution. (i) f(x) =e sin x

d
f(x) = esin x (sin x) = esin x cos x
dx
(ii) f(x) = sin (2x 3)
1 d cos(2x 3)
= . (sin (2x + 3)) =
2 sin (2x 3) dx sin (2x 3)
x
(iii) f(x) =
1 x2
(1 x 2 ) x(2x) 1 x2
f(x) = =
(1 x 2 )2 (1 x 2 )2
(iv)
f(x) = x sin x
f(x) = x. cos x + sin x
Solved Example # 3 If f(x) = sin (x + tanx) then find value of f(0).
Solution. f(x) = cos (x + tanx) (1 + sec 2x) f(0) = 2
Self Practice Problems :
1. Find the derviative of following functions using first principle.
(i) f(x) = x sin x (ii) f(x) = sin2 x
Ans. (i) x cosx + sinx (ii) 2sin x cos x
lim f ( 5 t ) f (5 t )
2. Evaluate if f(5) = 7, then t0 Ans. 7.
2t
3. Differentiate the following functions
( x 1)
(i) (1 + 3x 2) (2x 3 1) (ii) (iii) 1 x 2
( x 2)( x 3)
1 x
(iv) (v) cos3 x sin x (vi) x ex sin x
1 x
sin x
(vii) (viii) n (sin x cos x)
1 cos x
x 2 2x 1 x 1
Ans. (i)6x (5x 3 + x 1) (ii) (iii) (iv) 1/ 2
2
( x 2) ( x 3) 2
1 x 2 (1 x ) (1 x )3 / 2
1 x cos x sin x
(v) cos4 x 3 cos2x sin2x (vi) ex ((sin x + cos x) x + sin x) (vii) sec2 (viii)
2 2 sin x cos x
B. Derivative Of Inverse Trigonometric Functions.

y = sin1 x y x = sin y
2 2
dx
= cos y
dy
dy 1 1 dy 1
= = = 1 < x < 1.
dx cos y 1 sin 2 y dx 1 x2

Note here that cos y 1 sin 2 y , rather cos y =
1 sin 2 y but for values of y , , cos y is always
2 2
positive and hence the result. similarly let us find derivative of other inverse trigonometric functions.
Let y = tan1x
x = tan y
dx dx
= sec2y = 1 + tan2 y dy
= 1 + x2
dy
dy 1
= (x R)
dx 1 x2

Also if y = sec1x y [0, ] x = secy
2
dx dy 1 dy 1
dy
= sec y tan y = =
dx x. tan y dx x sec 2 y 1

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1
sec y 1
dy x x2 1 dy 1
= = x ( , 1) (1, )
dx 1 dx | x | x2 1
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sec y 1
x x 2 1
results for the derivative of inverse trigonometric functions can be summarized as :
f(x) f(x)
1
sin1x ; |x| < 1

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1 x2
1
cos1x ; |x| < 1
1 x2
1
tan1x ; xR
1 x2
1
cot 1x ; xR
1 x2
1
sec1 x ; |x| > 1
| x | x2 1
1
cosec-1 x ; |x| > 1
| x | x2 1
Solved Example # 4 If f(x) = n (sin1 x 2) find f(x)
1 1 2x
Solution. f(x) = . . 2x =
1 2
(sin x ) 1 ( x 2 )12 (sin x ) 1 x 4
1 2

Solved Example # 5 1
If f(x) = 2x sec x cosec (x) then find f(2)
2x 1
Solution. f(x) = 2 sec1(x) 2 +
| x | x 1 | x | x2 1
2 1 4 5
f(2) = 2.sec1( 2) + + f(2) = + .
3 2 3 3 2 3
C. Methods Of Differentiation
1. Logrithmic Differentiation
The process of taking logarithm of the function first and then differentiate is called Logarithmic Differentiation.
It is useful if
(i) a function is the product or quotient of a number of functions OR
(ii) a function is of the form [f(x)] g(x) where f & g are both derivable,
dy
Solved Example # 6 If y = x x find
dx
1 dy 1 dy
Solution. n y = x n x . = x . + n x = x x (1 + n x)
y dx x dx
dy
Solved Example # 7 If y = (sin x) n x, find
dx
Solution. n y = n x . n (sin x)
1 dy 1 cos x dy n sin x
= n (sin x) + n x. = (sin x) n x cot x n x
y dx x sin x dx x
x1/ 2 (1 2x )2 / 3 dy
Solved Example # 8 If y= find
( 2 3 x )3 / 4 (3 4 x ) 4 / 5 dx
1 2 3 4
Solution. n y = n x + n (1 2x) n (2 3x) n (3 4x)
2 3 4 5
1 dy 1 4 9 16
y dx
= + 4 (2 3 x )
+ 5 (3 4 x )
2x 3(1 2 x)
dy 1 4 9 16
= y 2x 3 (1 2x ) 4(2 3 x ) 5 (3 4x )
dx
2. Implicit differentiation
If f(x, y) = 0, is an implicit function then in order to find dy/dx, we differentiate each term w.r.t. x regarding y as a
functions of x & then collect terms in dy/dx.
dy
Solved Example # 9 If x 3 + y3 = 3xy find
dx
Solution. Differentiation both sides w.r.t.x, we get
dy dy dy y x2
3x 2 + 3y2 = 3x + 3y = 2
dx dx dx y x
2
Note that above result holds only for points where y x 0
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dy
Solved Example # 10 If x y = ex y, then find
dx
Solution.
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Taking log on both sides

y n x = (x y) .........(i)
differentiating w.r.t x, we get
y
1 xy
y dy dy dy x dy
+ lnx =1 = =
n x )

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x dx dx dx 1 n x dx x (1
dy
Solved Example # 11 If x y + yx = 2 then find
dx
du dv
Solution. u+v=2 + =0
dx dx
where u = x y & v = yx
n u = y n x & n v = x n y
1 du y dy 1 dv x dy
= + n x & = n y + y
u dx x dx v dx dx
du y dy dv x dy
= x y n x & = yx n y y dx
dx x dx dx
x y y
y n y x .
y dy x dy dy x
x y n x + y x n y

x dx y dx = 0.
dx
=
y x
x n x y x .
y
Self Practice Problems
1 x
1. Differentiate the following functions : (i) y = sec1 (x 2) (ii) y = tan1
1 x
x
1 x
(iii) y = 1 (iv) y = ex (v) y = (ln x)x + (x)sin x
x
dy
2. Find if
dx
(i) y = cos (x + y) (ii) x 2/3 + y2/3 = a2/3 (iii) x = y n (x y)
dy n x
3. If x y = ex y, then prove that
= .
dx (1 n x )2
x a dy x
4. If = log , prove that =2 .
xy xy dx y
x
2 1 1 1 1
Ans. 1. (i) (ii) (iii) 1 n 1 x 1 x
x x4 1 1 x2 x
1 sin x
(v) n (nx ) nx (n x)x + x sinx cos x nx
x
(iv) x x. e x (nx + 1)
x
1/ 3
sin( x y ) y y( x y )
2. (i) (ii) (iii)
1 sin( x y ) x x( x y )
3. Differentiation using substitution
Following substitutions are normally used to sumplify these expression.
(i) x2 a2 x = a tan or a cot
(ii) a2 x2 x = a sin or a cos
(iii) x 2 a2 x = a sec or a cosec
xa
(iv) x = a cos
ax
1 x 2 1

Solved Example # 12 : Differentiate y = tan1 .
x


Solution. Let x = tan = tan1x ; ,
2 2
| sec | 1
y = tan1 [ |sec| = sec , ]
tan 2 2

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1 cos
y = tan1 y = tan1 tan

sin 2

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y= [tan1 (tanx) = x for x , ]
2 2 2
1 dy 1
y = tan1 x =
2 dx 2(1 x 2 )

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dy 1 x 1 x
Solved Example # 13 : Find where y = tan1

dx 1 x 1 x
Solution. x = cos
= cos1 (x) ; [0, ]

1 cos 1 cos 2 cos 2 sin
2 2
y = tan1 1 cos 1 cos y = tan1

2 cos 2 sin
2 2

1 tan
2
y = tan1 y=
1 tan 4 2
2
1 dy 1
y= cos1x =
4 2 dx 2 1 x2

Note that 1 cos = 2 cos but for 0, 2 , 2 cos = 2 cos
2 2 2 2

Also tan1 (tan x) = x for x , .
2 2
2x
Solved Example # 14 If f(x) = sin1 then find
1 x2
1
(i) f(2) (ii) f (iii) f(1)
2
Solution. x = tan

= tan1(x) ; << y = sin1 (sin 2)
2 2
2
x 1
2 2 1 1 x2
2 2 tan x x 1 2
1 1 x 1
2 tan x 1 x 1 1 x2
y = 2 2
2 2 f(x) = f(x) =
( 2 tan 1 x ) x 1 2
( 2 ) 2 x 1
1 x 2
2
2 1 8
(i) f(2) = (ii) f = (iii) f(1+) = 1 & f(1) = + 1
5 2
5
f(1) does not exist.
Aliter Above problem can also be solved without any substution also, but in a little tedious way.
2x
f(x) = sin1
1 x2
1 2{(1 x 2 ) 2x 2 }
f(x) = .
4x 2 (1 x 2 )2
1
(1 x 2 )2
(1 x 2 ) 2(1 x 2 )
= 2 2 .
(1 x ) (1 x 2 )2
2
| x|1
2 (1 x ) 2 1 x 2
f(x) = . thus f(x) = 2
(1 x 2 ) | 1 x2 | |x|1
1 x 2
dy 1 y2
Solved Example # 15 If 1 x 2 + 1 y 2 = a(x y), then prove that = .
dx 1 x2
Solution. Put x = sin = sin1 (x)
y = sin = sin1 (y)
cos + cos = a (sin sin)
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2cos cos = 2a cos sin cot =a
2 2 2 2 2
= 2 cot 1 (a) sin1 x sin1 y = 2 cot 1(a)
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differentiating w.r.t to x.
1 1 dy dy 1 y2
=0 =
1 x2 1 y2 dx dx 1 x2
Aliter Using implicit differention.

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x y dy dy
2 = a 1
1 x 1 y2 dx dx
x
a
y dy x dy 1 x2
a =a+ 2 =
1 y2 dx 1 x dx
a
y
1 y2
1 x2 1 y2 x

dy xy 1 x2
=
dx 1 x 2 1 y2 y

xy 1 y 2

dy (1 x 2 ) (1 x 2 )(1 y 2 ) x 2 xy 1 y2 1 (1 x 2 )(1 y 2 ) xy 1 y2
= . = .
dx (1 x 2 )(1 y 2 ) (1 y 2 ) xy y 2 1 x2 1 (1 x 2 )(1 y 2 ) xy 1 x2
dy 1 y2
= Hence proved
dx 1 x2
dy dy / d
4. Parametric Differentiation If y = f() & x = g() where is a parameter, then dx dx / d .
dy
Solved Example # 16 If x= a cos3t and y = a sin3t. Find
dx
2
dy dy / dt 3a sin t cos t
= = = tan t
dx dx / dt 3a cos 2 t sin t
dy
Solved Example # 17 If y = a cos t and x = a (t sint) find the value of at t = .
dx 2
dy a sin t )
= a(1 cos t )
dx
dy
dx
= 1.
t
2
5. Derivative of one function with respect to another
dy dy / dx f ' (x)
Let y = f(x); z = g(x) then .
dz dz / dx g'(x)
Solved Example # 18
Find derivative of y = n x with respect to z = ex.
dy dy / dx 1
= =
dz dz / dx xex
Self Practice Problems :
dy
1. Find when
dx
(i) x = a (cos t + t sin t) & y = a (sin t t cos t)
1 t 2 2t
(ii) x = a
2 & y=b.
1 t 1 t2
(t 2 1)b
Ans. (i) tan t (ii)
2at
x2
dy 2xa 2
2. If y = sin1 4 4 then prove that = .
x a dx x4 a4
2x dy 2
3. If y = tan1 2
then prove that = (| x | 1)
1 x dx 1 x2
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du 1 u2
4. If u = sin (m cos1x) and v = cos (m sin1 x) then prove that = .
dv 1 v 2
D. Derivatives of Higher Order
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Let a function y = f(x) be defined on an open interval (a, b). Its derivative, if it exists on (a, b) is a certain function
f (x) [or (dy/dx) or y ] & is called the first derivative of y w. r. t. x.
If it happens that the first derivative has a derivative on (a, b) then this derivative is called the second derivative of
y w. r. t. x & is denoted by f (x) or (d2y/dx 2) or y .
d3 y d d2 y

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Similarly, the 3 order derivative of y w. r. t. x, if it exists, is defined by 3 dx 2 It is also denoted by f (x)
rd
dx dx
or y .
Solved Example # 19
If y = x 3 n x then yand y
1
Solution. y = 3x 2 n x + x 3
x
y = 3x 2 n x + x 2
1
y = 6x n x + 3x 2 . + 2x
x
y = 6x n x + 5x
y = 6 n x + 11
x
1
Solved Example # 20 If y = then find y(1)
x
Solution.
n y = x n x when x = 1 y=1
y
y
= (1 + n x) y = y (1 + n x) ......(i)
again diff. w.r.t. to x,
1 y
y = y(1 + n x) y . y = y (1 + ln x) 2 (using (i)) y(1) = 0
x x
It must be carefully noted that in case of parametric functions
dy dy / dt d2 y d2 y / dt 2 d2 y d dy / dt
although = but rather =
2 2 2
dx dx / dt dx dx / dt dx 2 dx dx / dt
which on applying chain rule can be resolved as
dx d2 y dy d2 x
. .
dt dt 2 dt dt 2
d2 y d dy / dt dt d2 y dt
2 = dt
. 2 = 2 .
dx dx / dt dx dx dx dx

dt
2
dx d y dy d x 2
. 2 .
2
d y dt dt dt dt 2
= 3
dx 2 dx

dt
d2 y
Solved Example # 21 If x = t + 1 and y = t 2 + t 3 then find .
dx 2
dy dx
Solution. = 2t + 3t 2 ; =1
dt dt
dy d2 y d dt
= 2t + 3t 2 2 = (2t + 3t 2) .
dx dx dt dx
2
d y
= 2 + 6t.
dx 2
d2 y
Solved Example # 22 If x = 2 cos t cos 2t and y = 2 sin t sin 2t then find value of 2 at t = .
dx 2
dy dx
Solution. = 2 cos t 2 cos 2t = 2 sin 2t 2 sin t
dt dt
3t t
2 sin . sin
dy cos t cos 2t 2 2
= = .
dx sin 2t sin t 3t t
2 cos . sin
2 2
dy 3t d2 y d 3t d2 y d 3t dt
= tan 2 = dx
tan 2 = tan .
dx 2 dx 2 dx dt 2 dx

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3 3t
. sec 2 d2 y
2 d2 y 2 3
= =
dx 2 2 (sin 2t sin t ) dx 2 t 2
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Solved Example # 23 Find second order derivative of2y= sin x with respect to z = e x.
dy dy / dx cos x
Solution. = =
dz dz / dx ex
d2 y d cos x d2 y d cos x dx
= x = .

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dz 2 dz e dx 2 dx e x dz
e x sin x cos xex 1
= . x
x 2
(e ) e
d2 y (sin x cos x )
=
dz 2
e2x
Solved Example # 24: y = f(x) and x =g(y) are inverse functions of each other than express g(y) and g(y) in terms of
derivative of f(x).
dy dx
Solution. = f(x) and = g(y)
dx dy
1
g(y) = ...........(i) again differentiating w.r.t. to y
f ( x )
d 1
g(y) =
dy f ( x )
d 1 dx
= .
dx f ( x ) dy
f ( x )
= . g(y)
f ( x )2
f ( x )
g(y) = .........(ii) which can also be remembered as
f ( x )3
d2 y
2
d x
dx 2
3 .
2 =
dy dy

dx
Solved Example # 25 y = sin (sinx) then prove that y + (tanx) y + y cos2x = 0
Solution. Such expression can be easily proved using implict differention.
y = cos (sin x) cos x sec x.y = cos (sin x)
again differentiating w.r.t x, we can get
secx y + y sec x tan x = sin (sin x) cos x
tanx y = y . cos2 x y +(tanx) y + y cos2x = 0
Self Practice Problems :
n x d2 y 2n x 3
1. If y = then find Ans.
x dx 2
x3
2. Prove that y = x + tan x satisfies the differentiation equation
d2 y
cos2 x 2y + 2x = 0.
dx 2
d2 y sec 3
3. If x = a (cos + sin ) and y = a(sin cos) then find . Ans.
dx 2 a
x sin x cos x
4. Find second derivative of nx with respect to sin x. Ans.
x 2 cos 3 x
5. if y = e x (A cos x + B sin x), prove that
d2 y dy
2 + 2 . dx + 2y = 0.
dx
d2 y dy
Solved Example # 26 If y = (tan1x)2 then prove that (1 + x 2)2 + 2x (1 + x 2) = 2.
dx 2 dx
Solution.
dy 2 tan 1 x
=
dx 1 x2
dy d2 y dy 2
(1 + x 2) = 2tan1 (x) (1 + x 2) + 2x =
dx dx 2 dx (1 x 2 )
d2 y dy
(1 + x 2) + 2x (1 + x 2) =2
dx 2 dx
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f(x) g( x ) h( x )
l( x ) m( x ) n( x )
11. If F(x) = , where f, g, h, l, m, n, u, v, w are differentiable functions of x then F (x)
u( x ) v( x ) w( x )
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f ' ( x ) g' ( x ) h' ( x ) f(x) g( x ) h( x ) f(x) g( x ) h( x)

l( x ) m( x ) n( x ) l' ( x ) m' ( x ) n' ( x ) l( x ) m( x) n( x)
= + +
u( x ) v( x ) w( x) u( x ) v( x ) w( x ) u' ( x ) v' ( x) w ' ( x )

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12. L Hospitals Rule:
If f(x) & g(x) are functions of x such that:

(i) Limit f(x) = 0 = Limit g(x) OR Limit f(x) = = Limit

x a x a x a x a g(x) &

(ii) Both f(x) & g(x) are continuous at x = a &

(iii) Both f(x) & g(x) are differentiable at x = a &
(iv) Both f (x) & g (x) are continuous at x = a, Then
f( x) f ' (x) f " (x)
Limit Limit Limit
x a g( x ) = x a g' ( x ) = x a g" ( x ) & so on till indeterminant form vanishes
________________________________________________________________________________________________

QUESTION BANK ON METHOD OF

DIFFERENTIATION
Select the correct alternative : (Only one is correct)
1
Q.1 If g is the inverse of f & f (x) = then g (x) =
1 x 5
1 1
(A) 1 + [g(x)]5 (B) 5
(C) (D) none
1 [g(x)] 1 [g(x)]5
n e2 d 2y
Q.2 If y = tan1 x
+ tan1 3 2 n x then =
nex 2 dx 2
1 6 n x
(A) 2 (B) 1 (C) 0 (D) 1
3x 4 dy
Q.3 If y = f & f (x) = tan x2 then =
5x 6 dx
2
3x 4 1 3 tan x 2 4
(A) tan x3 (B) 2 tan . (C) f tan x2 (D) none
5x 6 (5x 6)2 5 tan x 2 6
dy 1
Q.4 If y = sin1 x 1 x x 1 x 2 & = + p, then p =
dx 2 x (1 x)
(A) 0 (B) sin1 x (C) sin1 x (D) none of these
2x 1 dy
Q.5 If y = f 2 & f (x) = sin x then =
x 1 dx

(A)
1 x x2 2x 1
sin 2 (B)

2 1 x x2 sin 2x 1 (C) 1 x x 2
sin
2x 1
(D) none
2 2 2
1 x 2 x 1
1 x
2
2
x 1 1 x 2 x 2 1

x10
Q.6 Let g is the inverse function of f & f (x) = . If g(2) = a then g (2) is equal to
1 x 2

5 1 a2 a 10 1 a 10
(A) (B) (C) (D)
210 a 10 1 a2 a2
dy
Q.7 If sin (xy) + cos (xy) = 0 then =
dx
y y x x
(A) (B) (C) (D) y
x x y
2x dy
Q.8 If y = sin1 2 then dx is :
1x x 2
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2 2 2
(A) (B) (C) (D) none
5 5 5
1
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1
Q.9 The derivative of sec1 2
w.r.t. 1 x 2 at x = is :
2x 1 2
(A) 4 (B) 1/4 (C) 1 (D) none
d d 2y
If y2 = P(x), is a polynomial of degree 3, then 2 y 3 . 2 equals :

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Q.10
dx dx
(A) P (x) + P (x) (B) P (x) . P (x) (C) P (x) . P (x) (D) a constant
Q.11 Let f(x) be a quadratic expression which is positive for all real x . If g(x) = f(x) + f (x) + f (x), then for any real
x, which one is correct .
(A) g(x) < 0 (B) g(x) > 0 (C) g(x) = 0 (D) g(x) 0
dy
Q.12 If xp . yq = (x + y)p + q then is :
dx
(A) independent of p but dependent on q (B) dependent on p but independent of q
(C) dependent on both p & q (D) independent of p & q both .
g (x) . cos x1 if x 0
Q.13 Let f(x) = where g(x) is an even function differentiable at x = 0, passing through the
0 if x 0
origin . Then f (0) :(A) is equal to 1 (B) is equal to 0 (C) is equal to 2 (D) does not exist
1 1 1 dy np
Q.14 If y = nm p m + mn p n + mp n p then at e m is equal to:
1x x 1x x 1x x dx
(A) emnp (B) emn/p (C) enp/m (D) none
log sin 2 x cos x
Q.15 Lim
x0 x has the value equal to
log 2 x cos
sin
2
2
(A) 1 (B) 2 (C) 4 (D) none of these

Q.16 If f is differentiable in (0, 6) & f (4) = 5 then Limit

f (4) f x c h=
2

x2
2x
(A) 5 (B) 5/4 (C) 10 (D) 20
Q.17 Let l = xLim xm (ln x)n where m, n N then :
0
(A) l is independent of m and n (B) l is independent of m and depends on m
(C) l is independent of n and dependent on m (D) l is dependent on both m and n
cos x x 1
f (x)
Q.18 Let f(x) = 2 sin x x 2 2x . Then Limit
x0 =
x
tan x x 1
(A) 2 (B) 2 (C) 1 (D) 1
cos x sin x cos x

Q.19 Let f(x) = cos 2x sin 2x 2 cos 2x then f =
2
cos 3x sin 3x 3 cos 3x
(A) 0 (B) 12 (C) 4 (D) 12
Q.20 People living at Mars, instead of the usual definition of derivative D f(x), define a new kind of derivative, D*f(x) by
the formula
f 2 (x h) f 2 (x)
D*f(x) = Limit where f(x) means [f(x)]2. If f(x) = x lnx then
h 0 h
D * f ( x ) x e has the value
(A) e (B) 2e (C) 4e (D) none
f (x) g (x)
Q.21 If f(4) = g(4) = 2 ; f (4) = 9 ; g (4) = 6 then Limit
x4 is equal to :
x 2
3
(A) 3 2 (B) (C) 0 (D) none
2
f (x 3h) f (x 2h)
Q.22 If f(x) is a differentiable function of x then Limit
h0 =
h
(A) f (x) (B) 5f (x) (C) 0 (D) none

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d 2x
Q.23 If y = x + ex then is :
dy 2
ex ex 1
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(A) ex (B) 3 (C) 2 (D)

x 3
1 e x
1 e x
1 e
d 2y
Q.24 If x2y + y3 = 2 then the value of at the point (1, 1) is :
dx 2

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3 3 5
(A) (B) (C) (D) none
4 8 12
g (x) . f (a ) g (a ) . f (x)
Q.25 If f(a) = 2, f (a) = 1, g(a) = 1, g (a) = 2 then the value of Limit xa xa
is:
(A) 5 (B) 1/5 (C) 5 (D) none
Q.26 If f is twice differentiable such that f (x) f (x), f (x) g(x)
2 2
h (x) f (x) g(x) and
h (0) 2 , h (1) 4
then the equation y = h(x) represents :
(A) a curve of degree 2 (B) a curve passing through the origin
(C) a straight line with slope 2 (D) a straight line with y intercept equal to 2.
RS 1 UV 1 RS UV w.r.t.
Q.27 The derivative of the function, f(x)=cos-1
T 13 W
(2 cos x 3 sin x) +sin1
13 T(2 cos x 3 sin x)
W
3
1 x 2 at x = is :
4
3 5 10
(A) (B) (C) (D) 0
2 2 3
Q.28 Let x
f(x) be a polynomial in x . Then the second derivative of f(e ), is :
(A) f (ex) . ex + f (ex) (B) f (ex) . e2x + f (ex) . e2x
(C) x
f (e ) e 2x (D) f (ex) . e2x + f (ex) . ex
1
Q.29 The solution set of f (x) > g (x), where f(x) = (52x + 1) & g(x) = 5x + 4x (ln 5) is :
2
(A) x > 1 (B) 0 < x < 1 (C) x 0 (D) x > 0
x2 1 x2 1 dy
Q.30 If y = sin1 2 + sec1
2 , x > 1 then is equal to :
x 1 x 1 dx
x x2
(A) 4 (B) 4 (C) 0 (D) 1
x 1 x 1
x x x x x x dy
Q.31 If y = ...... then =
a b a b a b dx
a b a b
(A) (B) (C) (D)
ab 2 ay ab 2 by ab 2 by ab 2 ay
Q.32 Let f (x) be a polynomial function of second degree. If f (1) = f (1) and a, b, c are in A.P., then f '(a), f '(b) and
f '(c) are in
(A) G.P. (B) H.P. (C) A.G.P. (D) A.P.
y y1 y2
Q.33 If y = sin mx then the value of y 3 y4 y 5 (where subscripts of y shows the order of derivatiive) is:
y6 y7 y8
(A) independent of x but dependent on m (B) dependent of x but independent of m
(C) dependent on both m & x (D) independent of m & x .
y
Q.34 If x2 + y2 = R2 (R > 0) then k = where k in terms of R alone is equal to
2 3
1 y
1 1 2 2
(A) 2 (B) (C) (D) 2
R R R R
Q.35 If f & g are differentiable functions such that g (a) = 2 & g(a) = b and if fog is an identity function then f (b)
has the value equal to :
(A) 2/3 (B) 1 (C) 0 (D) 1/2
x3
Q.36 Given f(x) = + x2 sin 1.5 a x sin a . sin 2a 5 arc sin (a2 8a + 17) then :
3
(A) f(x) is not defined at x = sin 8 (B) f (sin 8) > 0
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(C) f (x) is not defined at x = sin 8 (D) f (sin 8) < 0
Q.37 A function f, defined for all positive real numbers, satisfies the equation f(x2) = x3 for every x > 0 . Then the value
of f (4) =
(A) 12 (B) 3 (C) 3/2 (D) cannot be determined
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Q.38 Given : f(x) = 4x3 6x2 cos 2a + 3x sin 2a . sin 6a + n 2 a a 2 then :

(A) f(x) is not defined at x = 1/2 (B) f (1/2) < 0
(C) f (x) is not defined at x = 1/2 (D) f (1/2) > 0

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d 2y dy
Q.39 If y = (A + Bx) + (m emx 1)2 ex then 2 2m + m2y is equal to :
dx dx
(A) e x (B) emx (C) emx (D) e(1 m) x
ax bx
Q.40 Suppose f (x) = e + e , where a b, and that f '' (x) 2 f ' (x) 15 f (x) = 0 for all x. Then the product ab is
equal to
(A) 25 (B) 9 (C) 15 (D) 9
Q.41 Let h (x) be differentiable for all x and let f (x) = (kx + ex) h(x) where k is some constant. If h (0) = 5, h ' (0) =
2 and f ' (0) = 18 then the value of k is equal to
(A) 5 (B) 4 (C) 3 (D) 2.2
Q.42 Let ef(x) = ln x . If g(x) is the inverse function of f(x) then g (x) equals to :
(A) ex (B) ex + x (C) e ( x ex ) (D) e(x + ln x)
dy
Q.43 The equation y2 e xy = 9e 3 x2 defines y as a differentiable function of x. The value of for
dx
x = 1 and y = 3 is
15 9
(A) (B) (C) 3 (D) 15
2 5
Q.44 Let f(x) = x x and g(x) = x then :
x xx

(A) f (1) = 1 and g (1) = 2 (B) f (1) = 2 and g (1) = 1
(C) f (1) = 1 and g (1) = 0 (D) f (1) = 1 and g (1) = 1
Q.45 The function f(x) = ex + x, being differentiable and one to one, has a differentiable inverse f1(x). The value of
d 1
(f ) at the point f(l n2) is
dx
1 1 1
(A) (B) (C) (D) none
n2 3 4
log sin|x| cos3 x
Q.46 If f (x) = for |x| < x0
3 x 3
log sin|3x| cos
2
=4 for x = 0

then, the number of points of discontinuity of f in , is
3 3
(A) 0 (B) 3 (C) 2 (D) 4
(a x) a x (b x) x b dy
Q.47 If y = then wherever it is defined is equal to :
a x xb dx
x (a b) 2 x (a b) (a b) 2 x (a b)
(A) (B) (C) (D)
(a x) (x b) 2 (a x) (x b) 2 (a x) (x b) 2 (a x) (x b)
2
d y dy
Q.48 If y is a function of x then 2 +y = 0 . If x is a function of y then the equation becomes :
dx dx
3 2 2
d2 x dx d2 x dx d2 x dx d2 x dx
(A) + x = 0 (B) +y =0 (C) y =0 (D) x =0
d y2 dy d y2 dy d y2 dy d y2 dy
Q.49 A function f (x) satisfies the condition, f (x) = f (x) + f (x) + f (x) + ...... where f (x) is a differentiable
function indefinitely and dash denotes the order of derivative . If f (0) = 1, then f (x) is :
(A) ex/2 (B) ex (C) e2x (D) e4x
cos 6x 6 cos 4 x 15 cos 2 x 10 dy
Q.50 If y = , then =
cos 5x 5 cos 3x 10 cos x dx
(A) 2 sinx + cosx (B) 2sinx (C) cos2x (D) sin2x
3
d 2 x dy d2y
Q.51 If + 2 = K then the value of K is equal to
dy 2 dx dx
(A) 1 (B) 1 (C) 2 (D) 0
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1
1 1

Q.52 If f(x) = 2 sin 1 x sin 2 x (1 x) where x 0 , then f ' (x) has the value equal to
2
2 2
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(A) x (1 x) (B) zero (C) x (1 x) (D)

1

x2 if x 0
e
Q.53 Let y = f(x) =

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0 if x 0
Then which of the following can best represent the graph of y = f(x) ?

(A) (B) (C) (D)

1 1 1
m n m n m n m n
Q.54 Diffrential coefficient of x
m n
. x n . x m w.r.t. x is


(A) 1 (B) 0 (C) 1 (D) xmn

Q.55 Let f (x) be diffrentiable at x = h then Lim

b g
x h f ( x) 2 h f ( h)
is equal to
x h xh
(A) f(h) + 2hf '(h) (B) 2 f(h) + hf '(h) (C) hf(h) + 2f '(h) (D) hf(h) 2f '(h)
d 3y
Q.56 If y = at2 + 2bt + c and t = ax2 + 2bx + c, then equals
dx 3
2
(A) 24 a (at + b) (B) 24 a (ax + b)2 (C) 24 a (at + b)2 (D) 24 a2 (ax + b)
1 x x
Q.57 Limit a arc tan b arc tan has the value equal to
x 0 x x a b
ab (a 2 b 2 ) a 2 b2
(A) (B) 0 (C) (D)
3 6a 2 b 2 3 a 2 b2
x
Q.58 Let f (x) be defined for all x > 0 & be continuous. Let f(x) satisfy f f ( x ) f ( y) for all x, y & f(e) = 1.
y
Then :
1
(A) f(x) is bounded (B) f 0 as x 0 (C) x.f(x)1 as x 0 (D) f(x) = ln x
x
Q.59 Suppose the function f (x) f (2x) has the derivative 5 at x = 1 and derivative 7 at x = 2. The derivative of the
function f (x) f (4x) at x = 1, has the value equal to
(A) 19 (B) 9 (C) 17 (D) 14
4 2
x x 1 dy
Q.60 If y = 2 and = ax + b then the value of a + b is equal to
x 3x 1 dx
5 5 5 5
(A) cot (B) cot (C) tan (D) tan
8 12 12 8
Q.61 Suppose that h (x) = f (x)g(x) and F(x) = f g ( x ) , where f (2) = 3 ; g(2) = 5 ; g'(2) = 4 ;
f '(2) = 2 and f '(5) = 11, then
(A) F'(2) = 11 h'(2) (B) F'(2) = 22h'(2) (C) F'(2) = 44 h'(2) (D) none
Q.62 Let f (x) = x3 + 8x + 3
which one of the properties of the derivative enables you to conclude that f (x) has an inverse?
(A) f ' (x) is a polynomial of even degree. (B) f ' (x) is self inverse.
(C) domain of f ' (x) is the range of f ' (x). (D) f ' (x) is always positive.
Q.63 Which one of the following statements is NOT CORRECT ?
(A) The derivative of a diffrentiable periodic function is a periodic function with the same period.
(B) If f (x) and g (x) both are defined on the entire number line and are aperiodic then the function F(x) = f (x) .
g (x) can not be periodic.
(C) Derivative of an even differentiable function is an odd function and derivative of an odd differentiable function
is an even function.
(D) Every function f (x) can be represented as the sum of an even and an odd function
Select the correct alternatives : (More than one are correct)
dy
Q.64 If y = tan x tan 2x tan 3x then has the value equal to :
dx
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2 of These Packages
2 & Learn by Video2 Tutorials on www.MathsBySuhag.com
(A) 3 sec 3x tan x tan 2x + sec x tan 2x tan 3x + 2 sec 2x tan 3x tan x
(B) 2y (cosec 2x + 2 cosec 4x + 3 cosec 6x)
(C) 3 sec2 3x 2 sec2 2x sec2 x (D) sec2 x + 2 sec2 2x + 3 sec2 3x
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x x dy
Q.65 If y = e e then equals
dx
e x e x e x e x 1 1
(A) (B) (C) y2 4 (D) y2 4

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2 x 2x 2 x 2 x
2 dy 2 2 2 2
Q.66 If y = xx then = (A) 2 ln x . xx (B) (2 ln x + 1). xx (C) (2 ln x + 1). x x 1 (D) x x 1 . ln ex2
dx
dy
Q.67 Let y = x x x ...... then =
dx
1 x 1 y
(A) (B) (C) (D)
2y 1 x 2y 1 4x 2x y
dy
Q.68 If 2x + 2y = 2x + y then has the value equal to :
dx

(A) x
2y
(B)
1
(C) 1 2 y
2x 1 2y
(D) y x

2 1 2x 2 2 1
Q.69 The functions u = ex sin x ; v = ex cos x satisfy the equation :
du dv d2u d 2v
(A) v u = u2 + v2 (B) = 2v (C) = 2u (D) none of these
dx dx dx2 dx 2
x 2 x 1
Q.70 Let f (x) = . x then :
x 1 1
(A) f (10) = 1 (B) f (3/2) = 1 (C) domain of f (x) is x 1 (D) none
Q.71 Two functions f & g have first & second derivatives at x = 0 & satisfy the relations,
2
f(0) = , f (0) = 2 g (0) = 4g (0) , g (0) = 5 f (0) = 6 f(0) = 3 then :
g(0)
f (x) 15
(A) if h(x) = then h (0) = (B) if k(x) = f(x) . g(x) sin x then k (0) = 2
g(x) 4
g (x) 1
(C) Limit
x0 = (D) none
f (x) 2
n ( n x ) dy
Q.72 If y = x ( n x ) , then is equal to :
dx
y y
(A)
x

n x n x 1 2 n x n n x (B)
x
(ln x)ln (ln x) (2 ln (ln x) + 1)
y 2 y n y
(C) ((ln x) + 2 ln (ln x)) (D) (2 ln (ln x) + 1)
x n x x n x

ANSWER KEY
Q.1 A Q.2 C Q.3 B Q.4 D Q.5 B
Q.6 B Q.7 B Q.8 C Q.9 A Q.10 C
Q.11 B Q.12 D Q.13 B Q.14 D Q.15 C
Q.16 D Q.17 A Q.18 B Q.19 C Q.20 C
Q.21 A Q.22 B Q.23 B Q.24 B Q.25 C
Q.26 C Q.27 C Q.28 D Q.29 D Q.30 C
Q.31 D Q.32 D Q.33 D Q.34 B Q.35 D
Q.36 D Q.37 B Q.38 D Q.39 A Q.40 C
Q.41 C Q.42 C Q.43 D Q.44 D Q.45 B
Q.46 C Q.47 B Q.48 C Q.49 A Q.50 B
Q.51 D Q.52 B Q.53 C Q.54 B Q.55 A
Q.56 D Q.57 D Q.58 D Q.59 A Q.60 B
Q.61 B Q.62 D Q.63 B
Q.64 A,B,C Q.65 A,C Q.66 C,D Q.67 A,C,D
Q.68 A,B,C,D Q.69 A,B,C Q.70 A,B Q.71 A,B,C Q.72 B,D

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EXERCISE -1
Part : (A) Only one correct option
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2
dy
1. If f(x) = 2x 2 1 and y = f(x ) then dx at x = 1 is
(A) 2 (B) 1 (C) 2 (D) 1
x2 dy
2. If y = x then =

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dx
2 2 x2 1 x2 1
(A) 2 n x. xx (B) (2 n x + 1). xx (C) (2 n x + 1). x (D) x . n ex 2
x
tan 1 sin
2
3. If f(x) = e , then f(0).
1 1
(A) (B) (C) 1 (D) 1
2 2
x dy
4. If y = then =
x dx
a
x
b
a .......... .....
a b a b
(A) (B) (C) (D)
ab 2 ay ab 2 by ab 2 by ab 2 ay
2
d2 F
5. Let f(x) = sin x; g(x) = x & h(x) = loge x & F(x) = h[g(f(x))] then is equal to:
dx 2
(A) 2 cosec3 x (B) 2 cot (x 2) 4x 2 cosec2 (x 2) (C) 2x cot x 2 (D) 2 cosec2 x
dy
6. If y = (1 + x) (1 + x 2) (1 + x 4) .....(1 + x 2n ), then at x = 0 is
dx
(A) 1 (B) 1 (C) 0 (D) 2n
dy 1
7. If y = sin1 x 1 x x 1 x 2 and = + p, then p =
dx 2 x (1 x)
1 1
(A) 0 (B) (C) sin1 x (D)
1 x 1 x2

y dy
8. If x 2 y 2 = et where t = sin1 then :
x y 2
2 dx

xy xy 2x y xy
(A) (B) (C) (D) 2x y
xy xy xy
2 2
x 1 x 1 dy
9. If y = sin1 2 + sec1 2 , x > 1 then is equal to:
x 1 x 1 dx
x x2
(A) (B) (C) 0 (D) 1
4
x 1 x4 1
t 1
10. The differential coefficient of sin1 2
w.r.t. cos1 is:
1 t 1 t2
1
(A) 1 (B) t (C) (D) none
1 t2
tan 1 x
11. Differentiation of 1
1 w.r.t. tan x is:
1 tan x
1 1 1
(A) (B) 1 (C) (D)
(1 tan 1 x)2
12.
1
1 tan x 1 tan1 x
Let f(x) be a polynomial in x. Then the second derivative of f(ex), is:
2

(A) f (ex). ex + f (ex) (B) f (ex). e2x + f (ex). e2x (C) f (ex) e2x (D) f (ex). e2x + f (ex). ex
f g h
f g h
13. If f(x), g(x), h(x) are polynomials in x of degree 2 and F(x) = , then F(x) is equal to
f g h
(A) 1 (B) 0 (C) 1 (D) f(x) . g(x) . h(x)
y y1 y 2
14. If y = sin mx then the value of 3 y 4 y 5 (where settings of y shows the order of derivative) is:
y
Successful People Replace the y 6 words
y 7 like;
y 8 "wish", "try" & "should" with "I Will". Ineffective People don't.
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(A) independent of x but dependent on m (B) dependent of x but independent of m
(C) dependent on both m & x (D) independent of m & x.
f (5 t ) f (5 t )
If f (5) = 7 then Limit
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15. t 0 =
2t
(A) 0 (B) 3.5 (C) 7 (D) 14
16. Let ef(x) = ln x. If g(x) is the inverse function of f(x) then g (x) equals to:
x
(A) ex (B) ex + x (C) e x e (D) ex + ln x

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dn
17. If u = ax + b then [f(ax + b)] is equal to:
dx n
dn dn dn dn
(A) [f(u)] (B) a n [f(u)] (C) an [f(u)] (D) an [f(u)]
du n du du n dx n
2x 1 dy
18. If y = f 2
& f (x) = sin x then =
x 1 dx

(A)
1 x x2
sin
2x 1
(B)
sin 2x 1
2 1 x x2
2
1 x 2 x 2 1 1 x
2
2
x 12

1 x x2 2x 1
(C) 2 sin (D) none
1 x 2 x 2 1

d 3 d 2y
19. If y2 = P(x), is a polynomial of degree 3, then 2 y . 2 equals:
dx dx
(A) P (x) + P (x) (B) P (x). P (x) (C) P (x). P (x) (D) a constant
Part : (B) May have more than one options correct
20. Two functions f & g have first & second derivatives at x = 0 & satisfy the relations,
2
f(0) = , f (0) = 2 g (0) = 4g (0), g (0) = 5 f (0) = 6 f(0) = 3 then:
g(0)
f (x) 15
(A) if h(x) = then h (0) = (B) if k(x) = f(x). g(x) sin x then k (0) = 2
g(x) 4
g (x) 1
(C) Limit
x 0 = (D) none
f (x) 2
fn 1 ( x ) d
21. If f n (x) = e for all n N and f o (x) = x, then {f (x)} is equal to:
dx n
d
(A) f n (x). {f (x)} (B) f n (x). f n 1 (x)
dx n 1
(C) f n (x). f n 1 (x)........ f 2 (x). f 1 (x) (D) none of these
22. If f is twice differentiable such that f(x) = f(x) and f(x) = g(x). If h(x) is a twice differentiable function such that
h(x) = [f(x)] 2 + [g(x)] 2 . If h(0) = 2, h(1) = 4, then the equation y = h(x) represents:
(A) a curve of degree 2 (B) a curve passing through the origin
(C) a straight line with slope 2 (D) a straight line with y intercept equal to 2.
x3
23. Given f(x) = + x 2 sin 1.5 a x sin a. sin 2a 5 sin1 (a2 8a + 17) then:
3
(A) f(x) = x 2 + 2x sin6 sin4 sin8 (B) f (sin 8) > 0
(C) f (x) is not defined at x = sin 8 (D) f (sin 8) < 0
24. 3 2
If f(x) = x + x f(1) + xf(3) for all x R then
(A) f(0) + f(2) = f(1) (B) f(0) + f(3) = 0 (C) f(1) + f(3) = f(2) (D) none of these
25. If f(x) = (ax + b) sin x + (cx + d) cos x, then the values of a, b, c and d such that f(x) = x cos x for all x are
(A) a = d = 1 (B) b = 0 (C) c = 0 (D) b = c

EXERCISE -2
dy d2 y
1. + n2 y = 0, where n2 = p2 + k2.
If y = A e kt cos (p t + c) then prove that 2 +2k
dt dt
Evaluate the following limits using L hospitale rule as otherwise
2. Limit log tan2 x (tan2 2 x)
x 0

( xa)4 ( xa)3 1 ( xa)4 ( xa)2 1

4 3 4
( xb ) ( xb ) 1 ( xb ) ( xb )2 1
3. If f (x) = then f (x) = . . Find the value of .
( xc )4 ( xc )3 1 ( xc )4 ( xc )2 1
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d3 y 8.b
4. If x = a t3 &y= b t 2, where t is a parameter, then prove that =
dx 3
27a 3 .t 7
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dy sina
5. If sin y = x sin (a + y), show that = 2 .
dx 1 2xcosax
F" f " g" 2c F f g
6.
If F(x) = f(x). g(x) & f (x). g (x) = c, prove that & .
F f g fg F f g

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7. If be a repeated root of a quadratic equation f(x) = 0 & A(x), B(x), C(x) be the polynomials of degree 3, 4 & 5
A( x ) B( x ) C( x )
A( ) B( ) C( )
respectively, then show that is divisible by f(x), where dash denotes the derivative.
A ' ( ) B' ( ) C' ( )
3/2
2
1 dy
dx 1 1
8. Show that R = 2 can be reduced to the form R2/3 = 2
2 .
d y 2 2
d y 3 d x 3
dx 2
dx 2

dy 2

Also show that, if x = a sin 2 (1 + cos 2) & y = a cos 2 (1 cos 2) then the value of R equals to
4 a cos 3.
9. Differentiate the following functions with respect to x.
sinx xcos x 1 cos x
(i) x 2. n x. ex (i) (iii) tan tan 1
x sinx cos x 1 cos x

Exercise # 1
1. A 2. C 3. A 4. D 5. D 6. B 7. D 8. B 9. C 10. A 11. C
12. D 13. B 14. D 15. C 16. C 17. C 18. B 19. C 20. ABC
21. AC 22. CD 23. AD 24. ABC 25. ABC
Exercise # 2
2. 1 3. 3
x2 1 x
9. (i) ex x (2 n x + 1 + x n x) (ii) (iii) sec2
(x sinx cos x)2 2 2

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