(ii) fffand g are

differentiable
ht)=f(x) -8(*), then h functions at and the

ie.f -gt) f (t)=f() g() -

function defined by
ie =
t) dr
-

(g ).
This, the derioative of the
iv) and g
f f ff and
difference of two functions is
are
differentiable
h)= f) 8), then F functions at
the
difference of their derioatives.
and if h is x
() f) &)+ g(*) =
the
function defined
f() by
iefgt) =ft)tr)+ gt).
d
f).
Thus, the derivative of the product two dr
first function derivative of
x
differentiable
of second function + second
functions
This known as product rule
is function derivative of first x

Extension of the
or
Leibnitz's rule. function.
product rule
If f. g and h are three
differentiable functions at x, then
ftdgth)) =ft) gt)h
Cx)+ft)htr)(gt)+ gkt)ftn.
In fact, this rule can be extended to any finite number of derivable
(w) If f and g are differentiable functions.
functions at x and h is the
function defined by
hx) = 8) 0, then h (x) = S)flt)-fl) g't)
glx))

ie d(f).3 gtr)f)-ftr). (gt)

g r ) # 0.
dxglx))
Thus, the derivative of the quotient of two
differentiable functions
deno. x derioative of num. -

num. x derivative of deno.

(denominator)
This is known as quotient rule.

6. ff is a differentiablefunction at x, then(G)= nifto)yn-1.f(t) where n is any positive

dx
integer.
7. Derivative of polynomials
Letflx) = ar" + ax-1 + a2-2 +.. + 4y-]k +4 be a polynomial function, where ag 4, az
n are fixed real numbers and n is a positive integer. Then the derivative of the polynomial f(t)

is given by

f(x) =
nax-1 + (n -

1)a,x2 + (n 2)azxS + -
. .

+ay-1
of the above results 1, 2, 3 and 5(), ().]
roof of this result is just the application

LLUSTRATIVE EXAMPLES
dmple 1. Find the derivatives ofthe following functions:
(ii) (ax2+
) 3x7-5r2+9
 dx

= 21 x6-10x.

dif. w.r.t. x, we get

(i) Let fCt)=+,
5

ra-)-)20)
- +5. r)=.1+5.(-1).r2
5dx dx

#0.

(n) Let f(x) = (ax2 +b)2 = a?t + 2abx2 + b2, diff. w.r.t. x, we get

P) a?4)+
dx
(2abr)
dx
+(
dx
)
ar)+2ab.) + ( )
=a2.4r3+ 2ab.2x+ 0 = 4a2x3+ 4abx

4ax (ax2 +b).

Example 2. Find the derivative of x" + ax!-l +a2x"=2 +... + a-lx +a", where a is
and n is any positive some fi
integer.
Solution. Let flx) =x" + ax"-1 +a2x"-2 +... +a'"-lx+a", diff. w.r.t. we
x, get
f(x) (x"
=

d +ax-1+a2x"-2 + . . +a"-1x +a")

()+a -1)+a2(-2)+.. +a"-1 ()+ dx dx dx

nx"-1 +a.(n -
1)x-2 +a2.(n-2)x"-3+ . . + a"-1.1+0
nx"-1 +(n + -1)ax1-2 (n -2)ax"-3+ + a"-1
Example
...

3. Find the
derivatives of the
following functions
()x4 (3- 4x 5)
(i)
Solution. (i) Let fx) =
x4
(3-4x-5) 3x4-4, diff. w.r.t. =

f)=3x-4)x-5- 4 (-9)x-10 x
x, we
get
-12x +
36x-10 -12x-5 (1
=
-3x).
i) Let f) =(x*=
= X+X+3x +
3x-1,
+3txx+
diff. w.r.t.
f(x)=3x4 (-3)x4 +3
+
X, We get
=3123
x 1
+3(-1)x4
 Va + _l
Example 4. fy= x+Prove that 2x dr +y=
= 2r.

Solution.Given y = vx +=xl/2,+ x-/2, diff. w.rt. x, we get

2 - 2x+=
2 +/=2y.

Example 5. Iy=*proe that 2xy

Solution.
Given y =
/2 Nax-2, diff. w.r.t.
x, we get

dxr
va
2a 2r2
2

Bample&.Fy1provethai x72-+?-3=0.
Solution. Given y =
2, diff. w.r.t. x, we get

5
-x 2+3x 2

2-y+-3= -x +3x -|3

5
x2 3x 2-x 2|+12-3=3-12412-3=0.

Example 7. Find the derivatives of the following functions:

3x+4
)(2+1)(x -2) (Gi)-7x+9
Uution.() Letf(x) = (x2+ 1) (x-2) = r3-2x2+ r-2, diff. w.r.t. x,weget

'o)= 32-2.2x!+1-0 312-4x +1.

 Altematively
w.r.t. x, we get
diff. w.a
(r-2),
fx)=(r2+)
1)
. 41
r - 1 ) . -
d 2
- 2 ) + ( r - 2 ) .

)
dx

(1-0) + (t-2)
(2r+0)
=(2+ 1)
= 3 r - 4r + 1,.
=2+1+2r4-4r

3 r+4diff. w.r.t. X, we get

() Let f)52-7x +9

(512-7x+9)(3r+4)-(3r +4)(52-7 +9
fla)= (52-71+9)2
(52-7x+9Y3 x1+0)-(x +4 (5 x 2x-7x1
+0)
(52-7x+9
3(5x2-7x +9)-(3x +4)(10x 7)
-

(5x2-7x+9)

152-21x+27-(30x +19x-28)55-40x-152
(5x4-7x+9) (5x2 7x +92

Example8. Find the

derivatives of thefollowing functions
a X +b
(i) px + qX +r

Solution. () Let ft) = , diff. w.rt. x, we get

f t ) = -)
Pt)- -
a")-(- a").t-a)
(r-a)2
(x-a(nx-l - 0) - (x" - a")(1-0)
(x-a)2
nx" nax"-l - " +a" (n -1)x" - nax"+ar+a.
(x-a)2 (r-a)2
(i) Let flx) = ax +b
- , diff. w.r.t. x,
px* +qx +r we get

f(x) pr+x+ r).ax +b)-(ax b). dx (px2 +gx +) +

(pr + gx + r)
px +qx +r(a.1+ 0)- (ax +
b)(p.2x + 9.l+0
(px +qx +r
=
a(px +qx +r)- (ax +
b)(2px +q)=apr2+ 2bpr +(by- aT
(px+qx+ (pr + qx +r
E x a m p l e
Find the derivativesof the following functions
9 ,F i n d .

+ b) (cr
+ d)2 ) (ax + by" (cx + dy'", n, m e N.
(ax
( Let fx) = (ar + b) (cx + d, diff. w.r.t. x, we get
Solutie

"(r) =(ax +)((Cx+ d) +(cx +dj? (ax +b)

dx dx (product rule)
= (ax + b).2(cx + d)'.c+ (cx + d2.(a.1 + 0)
( (ax + by") = n(ax + by-1.a, result 4)
dx
2c(ax + b) (cx + d) + a(cx + dy2
(2c(ax + b) + a(cx +d) (cx + d)
= (3acx + 2bc + ad) (cx + d).

(cx + d)", n, m
+ by" e N.
flt) (ax
=

Let

Differentiating
w.r.t. x, we get

Pa)= (0x +b"(cx+d") + (cx +dym,(ax +byr)

product rule)
= (ax + b".m(cx + d)y=.c+ (cx + dy".n(ax + byr-1.a (using result 4)
= (ax +by (cr +d) (mc(ax + b) +na(cx+ d)
(ax + b-1 (cx + d)y"=l (ac(m + n) x + (mbc + nad).

EXERCISE 11.2
Find the derivatives ofthefollowing (1 5) functions
to w.rt. x:

(i) 4 -2 (ii) ax3 + bx2+ cx + d (iv)2r+x

20 2
3. (i)xrs (5 +3x) (i) (3-6x).
4. () (2x +3) (5x2-7x +1) (i) (2x-7? (3x+5)3
5.)a-2) i)+52x2-1)

6. For thefunction f, givenby ft) = 2x2-3x +7, showthat f')=5f°().

7. f for the function off defined by ftx) = k?-5x+4.f 3) = 37, find the value of k.

8. Ify=x+ provethat x xy +2=0.

dx
Find the derivatives ofthefollowing (9 to 11) functions
1
4x +b ()+D(3r-1)
9.09 (i) cr+d
10.
x+5 ()? +bx + c

11. (i) P +qx+r (i) a+

.
bx +c
4X + b px + qx +r

Answers
10 2 (ii) 3ax2+ 2hx +c iv) 8rs+ 1

2. ()8x- 3. (i)-15xrt-6xrd (i) 1514 +24

(Gi) 1-
Derivative o fa b s o l u t e value function
= lxl= * , differentiating w.r.t. x,
Let we
get

dx

Thus, ) 0

TLLUSTRATIVE EXAMPLES
the follozwing fiunctions w.r.t.
Example
1, Differentiate x :

2x-3,x>
(i) (3x2 -5x + 1).
2
a n L e t f(x) = v2x - 3 = (2x - 3)'4, differentiating w.r.t. x, we get

Sol
f)= 2x-3)-1/2(2x
dx
-3)
= (2x -3)(2.1-0) = J - 3 2 1

5x + 1), differentiating w.r.t. x, we get

() Let y
=
(3x2 -

7.(312 5 x + 1)5 (32-5x 1)

+
=

5x + 1)6 (3.2x -

5.1 +0)
=
7(3x2 -

= 7(3x2 5x + 1 ) (6x - 5).

w.rt. x:
+1-
the following function
-

2+1 +x
Example 2. Diferentiate

+ 1 - v+1-,v+1-x
Solution. Let x+1+x vx+1 -x
y 2 +1+x
(+1)+ 2-2rv?+1 22+1-2rv21
(2+1)-x2

differentiating
w.r.t. x, we get
22 +1- 2x Vx2 +1,

2+1)-1/2.2x+ vx2+1.1
t+0-27
-2/2r-2
+V+14r-2. +1
=ir-2
Vx2+a ), prooe
that 2+a? ...()
Example 3. Ify =
(x +

Solution. Given y =(r+ vx2 +a2y"

w.r.t. x, we get
Differentiating
vr2+a2-1,(x +Vr2 +a2)
n (x +
dx
a?)r/2,2x
Vr+dy-11a?+
=n (a+
315
Differentiation
 3 2x
3. 0 23x+2 (212+4372> - (ii) 20x
Va?-(a2+r23/2
a(a-vad-r?)
(i) 2x + 2x3
4. (02a2-2 Ax(22 -3)
r-1 2r2-3

IMPLICIT IFFERENTIATION
be
function of x
a
fined by an equation such as
Ify
y 7d-531112 v2r-3 (i)
u is said to be defined explicitly in terms of r and we write y f(r) where

fa) = 74 -5x3 11r2 v2x -3.

However, if x and y are connected by an equation of the form
ty 3ys + 7 -8x2 +9 =0
-

cannot be expressed explicitly in terms of x. But, still the value of

ie fx,y) =0, then y
and there may exist one or more functions f connecting y with x so
depends upon that of x functions satisfying equation (i).
or there may not exist any of the
as to satisfy equation (17)
Consider the equations
Forexample: 0
(11)
+ -25 =

(iv)
r +y + 25 = 00
and a function
be expressed explicitly
in terms of x, but y is not
to be
In equation (i), y may considered
functions of y if y were
twofunctions of x (or two
we have
of x. Here, 2 which
defined by f(«) = 25-2 and f(r) =
-v25 -

independent variable) fi
and f
satisfy equation (ii).
real values of x that can satisfy it.
there are no
In equation (io), function of x (or x i an implicit
is an implicit the
and (i»), we say that y of y with regard to r (or
In cases (i), (ii) find the
derivative
Of course,
cases, w e
and in all such
differentiation.

function of y) the process

called implicit implicit
of with regard to y) by defines one
variable as an

equation that
derivative x
an differentiable.
wherever we differentiate implicitly that the function
is
we shall
assume

of another variable,
function

ILLUSTRATIVE EXAMPLES

100.
1. Findwhen
r2 + xy +y =

Example dx
100. w.r.t. x,
we get
+ y =
both sides
Solution. Given x2 + xy differentiating
function of x,
as a
Kegarding y 2)=2-y
=0 (* +
2x+ x+y.1
x
| + 2y dx
dy2x+y
x+2y dy ..()
+ y2/3 a23, find dx =

Example 2. f 23 we get
of x,
a2/3 as a
function

Solution. Given 12/3 + y2/3 =

x,
regarding y 1/3
w.r.t.
sides of (i)
Differentiating
both

7s*73 0- - 7
317
x - / 3 y - 1 / 3 , = 0

Differentiation
 EXERCISE T14

following
(1 to 4)
Fimdin the
(i) 2 + = 2 5 2. (i) xy c2
=

1. ()r-y=T

() r+ry+ Xy +y'=81.
(i) C
=y +y
4. ((r2+yP
find
at (1, 1).
5. If2/3+
y2/3= 2,

Answers 2. ()-2
. )1
32 +2xy +y
3 (- (i7)2+2xy +3y
4. - 4 3 - 4 x y 2 Gi 5. -1
.0)
47'y+4y-x

Derivatives of exponential and logarithmic functions

() Derivative of e is e" for all x e R.
Proof. Let ftx) = e", for all x e R.

By def., f(x) =
Lt * f) I+ e+h e -

h0 h h-0 h

h0 h h0 h

= e. 1
e

Thus, ) =
",for all x E R
(i) Derivative of
log * is , where x > 0.
Proof. Let ftx) log =
x, x> 0.

By def., flx) =
Lt log(x+ h)-logY log
h0
h Lt
h+0
- h

= Lt
log 1
h0 log 1+
>0

=1 log 1
 Thus, og x)
=
where x

Some
nportant deducti tions
of a', a > 0,
1. Derivative 1.
a*

d
(a) dx
(e log a)
dx
=
log a (x log a) e* log ". log a =

(x)
=
a log a.1 = a" log a.
Thus,
dx
a*) a log=
a, a > 0, a * 1.

of lo8a *, * > 0, a> 0, a z 1.

2. Derivative

00s ) ddlogx
log a (Base changing formula)

a'dg) = 1
loga dx loga x xloga

> 0, > 0, a + 1.
3) = x,loga
x a
Thus,
*0.
3. Derivative
of loglx|, x
Ologl|)= )=i
#0

Thus,(oglx|)=x*0.
> 0, a # 1.
Derivative of logalx|, x * 0, a
4. (Base charnging formula)

logIz|)=
1 lo1
log|x
log a dx
|)= loga loga
a*1.
x # 0, a > 0,
Thus,los|x|)=
dx
xloga
ILLUSTRATIVE EXAMPLES

functions
Diferentiate
the following
Example 1.

() 5* + logx w.r.t. x, we get

d i f f e r e n t i a t i n g

= 5 + logx,
Solution. () Let y
=
57log 5+
5 log 5 +
Difterentiation
319
 w.r.t. x, we get
B
differentiating

(i)Let y=
log8-8T8
8"(rlog 8-8)
r8" x6
dy (r)2

dx
8(xlog 8-8)

functions
Find the
derivatives of the following
Example 2.

+ vr< ad) -

(i) log (X+1+xV -1\

() log (r +1- Jx-1)
Let y =
log (r+ vr2-a2). differentiatino w.r.t.
Solution. () X, We
get
dy +2-r?)
dx r+ yr2-a2

+r-a2 l 1-y/2,2x 1
1 2-a +X=. 1

X+Vx4-a< 2-2 V2-2

(V
) Let y log +1+x
= - =log (Vx+1tv*-1 vx +1+ vt-1|
|+1-x -

x+1- vx -1 Vz+1+vr-1
(x+1)+(x(+1)-(x-)
-1) +2/x
= log
log2 2
=
log (r+ Vx -1), differentiating w.r.t. x, we get
dy
-T1**-y/2.2

X +Vr2-1 a-1+ 2-1+ 1

X+vr2-1?-1?-1
Example 3. If y eslog X+2x, prove that
=

x?2x (2x +3).

Solution. Given y e3log x +2x elog =
dx
=
2x =r3. e2Xx (:
Differentiating w.r.t. x, we get
=
dx .e2x .2 + e2x
.32 ?e2x =
(2x + 3).
Example If y 9o%3*,
4. =

show that
Solution. Given =
dx 2x.
y =
9log3 =

(32ylog3
y x =
32log3 * =
3og3*
dx
2 2x.
Example 5. (i) If ex +
e =
erty, prove
that e(e-1)
i) If ex + e dx
=
ex t+
y, prove that e(e -1)
1x -el -x
 (i) Given e + e = etty
tion. ..i)
entiating both sides of (i) w.r.t. x,
Differenti
regarding y as a function of x, we get
.1 + e. ay = el*v.1+

+e = e*V +etty dy
dr
(eV -e+y) e+y-e
dx

-e(e-1)dy e ( - 1) = -C-)
e"(e -1)

in Given e + =
e *, dividing both sides by et +
V, we get
eV+ = 1, diff. w.r.t. x

e. + - 1 ) =0 ev =dx -

that = 1 -

2
Example 6. Ify :
prove dx
...0)
=
Given y
Solution.
e* (-1))
- ) + e ( - 1 ) -(e*+e*)(e¥
-

dy (e-e)2

(--+"E.1-
(e-e2
(using ()
=1 -
-1
prove that dx 2x(x + 1)
Example 7. fy =
log| vx+
os|log+ 1)-;log x,.
Solution. Given y =
logr+-
2x-(+1) -1
2x(x+1)
2x(r+ 1)
y
1)+xy
+1 = 0.
that (x +

log (Vx2 +1 -x),

prove
Example 8. fy vr2
+1 =
...0
+1 - )
Solution. Given y V2+1 =log(Vr2 as a
function ot x, we get
y
w.r.t. x, regarding
both sides of (i)
Diterentiating

1 2i-T12?.1
2r+Vx2
+

y
r -V2+1
+r2+1 +1-r + 1
+1
(2+ 1 ) + x y
1y + 1 =0.
dy -1
xy +(r2 +
=

ix

321
Differentiation
 +...o prove tha,
* Vogx x(2y 1
xrtyiog X -

=
ylog
9. fy 1
Example
logx + ... o

lo8I
+ VIOgX+
Given y=
Solution.

= ylog+ y
y

= log1 + y.
w.r.t. X, we get
Differentiating

(2y-= * (2y -1)=1

2 dx

EXERCISE 11.5
to 3) functions w.r.t. x
Diferentiate the following (1
(i) e-x
1. 10+ -2log x
2 ) (it) ve, x>0.
3. () log (log x), x > 1 (i) log7 (og X), x > 1.
4 (91ff ) = 4 and f'(1) = 2, find the value of the derivative o
e of
x = 0. log fe
(i) If flx) = e g(x), g(0) = 2 and g'(0) = 1, then find f"(0).

Diferentiate the following (5 to 6) functions wrt. x:

5. () e +e . +e i) o g
r2+1
66. )
0 logtV-2|
6-P-2)
(i) rv1+ x +log (x+ v+1).

7. () If y = =
log ( +x*+) prove that (2 +
2+1 1)+Xy
ax =1.
(i) If y =
e2log x 3x, prove that
+

dx
x(2 + 3x) e =

8.Find when
dx

) xy +
xe-y +
ye =

Gi)-y = log|
9. If y
log x =x - y,
prove
Hint. ylogx =x -
that=-
dx
log x
(1+logx
y y (1
Answers +log x) =
*
Y
1. i) 10 log 10
1+l08X
+
e
2. () 3x23 (i) -ex
i) evV*
(eg7

4e 3. (i) x log X
i) 3
 5. ( e2xe +3xe +41e+5rte (n 5"(r 101+rlog5log r)-2r log r)

6.
(i) 2r2+1

v(xe - 1)
(i)
x(veT-v -1)

mULTIPLE CHOICE QUESTIONS

anse the correct ansverfrom thegivenfour options in questions (1 to
Choo

1. Itf(x) = (3+1)(2-) then f (1) is equal to

11
a) 5 (b) -5 (c) 6

thenat=1 is
2 Ify=+
(b) 1 (d) 0
(a) 1

3. Ify
-
then

) -1 )-1

4. Ify=log Ix<1, then

4x
() (
)1P 1-
Answers
4. (b)
1. (a) 2. (d) 3. (a)

FILL In THE BLANKS

(1 to 8):
Fill in the blanks in questions

X *1,thenf'()=
*****
... . *

r100
1. 1fa)=

2. Iffx)= thenf'(1)
********

2Vx
1, thenf" (1)
=.

+r+

3. Iff(x) =r100+ 99+

198 .

(r)is ***

(fof)
4Ffx) =2r+1, then the derivativeof s.

vr+ y =
1, 4x atthepoint
5.
Forthe curve 2*" is .
***************

function f ()
=

6. The derivative of the ************

of log10 is.1s
*
derivative
then the **

. Ifr>0, |X| **** *****

derivative oflog2
the
8. Ifx*0, then
Ditterentiation