SUCCESSIVE DIFFERENTIATION
Introduction to topic :
It is extension of differentiation of one variable function.
Weightage for university exam: 08 Marks
No. of lectures required to teach: 04 hrs
Definition of Successive differentiation:
Consider,
A one variable function,
y = f(x) (x is independent variable and y depends on x.)
Here if we make any change in x there will be a related change in y.
This change is called derivative of y w.r.t. x. denoted by f’(x) or y1 or y’ or
dy called first order derivative of y w.r.t. x.
dx
f”(x) = (f’(x))’ = d2y = y” = y2 is called second order derivative of y w.r.t x.
dx2
It gives rate of change in y1w.r.t. rate of change in x.
Similarly,
Third derivative of y is denoted by y3 or f ’’’(x) or d3y or y’’’ and
dx3
So on………………………..
(Above derivatives exist because, If y = f(x), then y1 = g(x), where g(x) is
some function of x depends on f(x)
for e.g. if y = sinx then y1 = cosx ,hence y2 = -sinx and so on………..)
Thus,
Derivatives of f(x) (or f) w.r.t. x are denoted by, f ’(x), f ”(x), f ”’(x),
………………., f (n)(x),……....
Above process is called successive differentiation of f (x) w.r.t. x and f ’,
f ”,f ’”,……..,f(n) are called successive derivatives of f.
f (n)(x) denotes nth derivative of f.
Notations:
Successive derivatives of y w.r.t. x are also denoted by,
1. y1,y2,y3,……….yn,……………. …………or
2. dy ,d2y ,d3y ,……dny,……………………. or
dx dx2 dx3 dxn
3. f ’(x), f ”(x), f ”’(x), ……., f (n)(x),……......or
4. y’, y’’, y”’, ………………y(n),……………or
5. Dy, D2y, D3y,……………., Dny,…………….
Where D denotes d .
dx
(1) Prepared by Mr.Zalak Patel
Lecturer, Mathematics
Value of nth derivative of y = f(x) at x =a is denoted by ,
dn y
f n(a) ,yn(a), or n
dx xa
(i.e. value can be obtained by just replacing x with a in f n(x).)
List of formulas (01) :
Sr Function nth derivative
no.
01 y = eax yn = an eax
02 y = bax yn = an bax (logeb)n
03 y = (ax + b)m (i) if m is integer greater than n or less than (-1)
then, yn = m(m-1)(m-2)…..(m-n+1) an (ax + b)m-n
(ii) if m is less then n then, yn = 0
(iii) if m = n then, yn = an n!
(iv) if m = -1 then , yn = (-1)n n! an
(ax + b) n+1
(v) if m = -2 then , yn = (-1)n (n+1)! an
(ax + b) n+1
04 y = log (ax +b) yn = (-1)n-1 (n-1)! an
(ax + b) n
Problems Based On Above Formulas :
Obtain 5th derivative of e2x.
1.
Obtain 3rd derivative of 35x.
2.
Obtain 4th derivative of (2x +3 )5
3.
Obtain 4th derivative of (2x +3 )3
4.
Obtain 4th derivative of (2x +3 )4
5.
Obtain 4th derivative of ___1___
6.
(2x +3)
nth derivatives of reciprocal of polynomials (nth derivatives of functions
which contain polynomials in denominators) :
Consider
y= ax +b or y = _____1____
cx2+dx+e cx2+dx+e
To find nth derivative of above kind function first obtain partial fractions of f(x)
or y.
To get partial fractions:
If y = __1_____ then first factorize cx2+dx+e.
cx2+dx+e
Let (fx +g). (hx +i) be factors then y = _____1_____
(fx +g).(hx +i)
Find A & B such that y = _ A__ + __ B__
fx +g hx +i
(2) Prepared by Mr.Zalak Patel
Lecturer, Mathematics
obtain nth derivatives of above fractions separately and add them, answer will give
nth derivative of y.
Note:
If polynomial in denominator is of higher Degree then we will have more
factors .(Do the same process for all the factors).
If y= ___1______ then use factors y = __ A__ + __ B __ + __ C__
(fx +g)2.(hx +i) (fx +g)2 hx + i fx + g
Problems Based On Above Formulas & notes :
Obtain nth derivatives of followings:
(1) __1 (2) ax + b (3) x (4) x2 (5) 8x
a2 - x2 cx + d (x-1)(x-2)(x-3) (x +2) (2x+3) (x+2)(x-2)2
(6) x log (x -1) (7) __1__ (8) ___1___ .
(x+1) x2+a2 x2+x+1
ASSIGNMENT (01) :
Obtain nth derivatives of followings:
(1) _a - x (2) 1 (3) _____x4____ (4) ___x__
a +x (x-1)2(x -2) (x-1) (x-2) a2 + x2
List of formulas (02) :
Sr Function nth derivative
no.
01 y = sin(ax + b) (i)yn = an sin(ax +b +n π/2)
(ii)if b =0 , a =1 then y = sinx & yn = sin(x +n π/2)
02 y = cos(ax + b) (i)yn = an cos(ax +b +n π/2)
(ii)if b =0, a =1 then y =cosx & yn = cos(x +n π/2)
ax
03 y = e sin( bx + c) (i)yn = rn e ax sin(bx + c + nӨ)
where r = (a2 + b2)1/2
Ө = tan-1(b/a)
04 y = eax cos( bx + c) (i)yn = rn e ax cos( bx + c + nӨ)
where r = (a2 + b2)1/2
Ө = tan-1(b/a)
(3) Prepared by Mr.Zalak Patel
Lecturer, Mathematics
Problems Based On Above Formulas :
1. Obtain 4th derivative of sin(3x+5).
2. Obtain 3rd derivative of e2x cos3x
Problems Based On Above Formulas :
Obtain nth derivatives of followings:
(1) sinx sin2x (2) sin2xcos3x (3) cos4x (4) e2xcosx sin22x
ASSIGNMENT (02) :
Obtain nth derivatives of followings:
(1) cosxcos2xcos3x (2) sin4x (3) e-xcos2x sinx
Some Problems (Problems Of Special Type) Based On Above All
(01& 02) formulas:
(1) For y = __x3_,
x2-1
dny 0 if n is even
Show that, n =
dx x 0 (-n) if n is odd integer greater than 1
(2) If y = cosh2x, show that
yn = 2n sinh2x, when n is odd.
= 2n cosh2x, when n is even.
(3) Find nth derivative of following:
-1 1 - x
2
-1 1 - x -1 2 x
(i) tan (ii) sin 2
(iii) cos (iv) tan-1x
2
1 x 1 x 1 x
(4) If u = sinnx + cosnx, show that
1
ur = nr 1 (-1) r sin2nx 2
where ur denotes the rth derivative of u with respect to x.
dn
(5) If I n = n
(xn logx),
dx
Prove that I n = n In-1+ (n-1)!,
Hence show that
1 1 1
I n = n! ( logx + 1+ ........ )
2 3 n
(4) Prepared by Mr.Zalak Patel
Lecturer, Mathematics
Leibnitz’s theorem(only statement):
If y = u .v,
where u & v are functions of x possessing derivatives of nth order then,
yn = nC0unv +nC1un-1v1 + nC2un-2v2+………..+ nCrun-rvr+.......+ nCnuvn
where, nCr = ____n!___
r!(n-r)!
Properties:
1) nCr = nCn-r
2) nC0 = 1 = nCn
3) nC1 = n = nCn-1
Note:
Generally we can take any function as u and any as v.( If y = u .v)
But take v as the function whose derivative becomes zero after some order.
Problems Based On Leibnitz’s theorem:
Obtain nth derivatives of followings:
(1) x3 logx (2) __xn __ (3) x2 ex cosx
x+1
ASSIGNMENT (03) :
Obtain nth derivatives of followings (using Leibnitz’s theorem):
(1) x2 logx (2) x2 ex (3) x tan-1x .
Solved Problems (Problems Of Special Type) Based On Leibnitz’s theorem:
(1) If y = sin (msin-1x)
Then prove, (1-x2)yn+2-(2n+1)xyn+1+(m2-n2)yn = 0
(2) If y = cot-1x,
Then prove, (1+x2)yn+2+2(n+1)xyn+1+n(n+1)yn = 0
(3) If y1/m + y-1/m = 2x
Then prove, (x2-1)yn+2+(2n+1)xyn+1+(n2-m2)yn = 0
ASSIGNMENT (04) :
n
-1
y x
(1) If cos = log then prove, x2yn+2+ (2n+1)xyn+1+2n2yn = 0
b n
(2) If y = (x2-1)n then prove, (x2-1)yn+2+2xyn+1-n(n+1)yn = 0
a x
(3) If y = tan-1 then prove, (a2+x2)yn+2+2(n+1)xyn+1+n(n+1)yn.
a x
(5) Prepared by Mr.Zalak Patel
Lecturer, Mathematics
