CHAPTER 7 SUCCESSIVE DIFFERENTIATION
CHAPTER 7 SUCCESSIVE DIFFERENTIATION
TOPICS: 1 . Successive differentiation-nth derivative of a function ? theorems. 2. Finding the nth derivative of the given function. 3. Leibnitz's theorem and its applications.
SUCCESSIVE DIFFERENTIATION
Let f be a differentiable function on an interval I. Then the derivative f is a function of x and
if f is differentiable at x, then the derivative of f at x is called second derivative of f at x. It is denoted by f(x) or f(2)(x).similarly, if f" is differentialble at x , then this derivative is called the 3rd derivative of f and it is denoted by f(3)(x). Proceeding in this way the nth derivative of f is the derivative of the function f(n-1)(x) and it the denoted by f(n)(x).
If y = f(x) then
f (n)(x)
is denoted by
dny dx n
or Dn y or y(n) or yn
and f (n) ( x) = lim f (n-1) ( x + h) - f (n-1) ( x)
h0
h
THEOREM
If f(x) = (ax + b)m, m R, ax + b > 0 and n N then f (n) (x) = m(m -1)(m - 2)...(m - n +1)(ax + b)m-n an
Note : If y = (ax + b)m then yn = m(m ? 1)(m ? 2) ...(m ? n + 1)(ax + b)m?n an.
COROLLARY If f(x) = (ax + b)m, m Z, m > 0, n N then (i) m < n f(n)(x) = 0, (ii) m = n f(n)(x) = n! an
(iii) m > n f(n)(x) = m! (ax + b)m-n an . (m - n)!
COROLLARY If f(x) is a polynomial function of degree less than n where n N then f(n)(x) = 0.
THEOREM
If f (x) = 1 ax + b
then f(n)(x) =
(-1)n (ax +
n !a n b)n+1
.(i.e.,
If y = 1 ax + b
yn =
(-1)n (ax +
n !a n b)n+1
)
THEOREM
If
f(x)
=
log
|ax
+
b|
and
n
N
then
f (n)(x)
=
(-1)n-1(n -1)!an (ax + b)n
.
i.e., y = log |ax+b|
yn
=
(-1)n-1(n -1)!an (ax + b)n
THEOREM
If
f(x)
=
sin(ax
+
b)
and
n
N
then
f (n)(x)
= an
sin
ax
+
b
+
n 2
.
THEOREM
If
f(x)
=
cos(ax
+
b)
and
n
N
then
f (n)(x)
=
an
cos
ax
+
b
+
n 2
.
THEOREM
If f(x) = eax+b and n N then f (n) (x) = aneax+b .
THEOREM If f(x) = cax+b, c > 0 and n N then f (n) (x) = ancax+b (log c)n .
THEOREM
If f(x) = eax sin(bx + c) and n N then f (n)(x) = rneax sin(bx + c + n) where a = r cos , b = r sin
and r =
a2
+
b2
,
=
Tan -1
b a
THEOREM
If f(x) = eax cos(bx + c) and n N then f (n) (x) = rneax cos(bx + c + n) where a = r cos , b = r sin
and r =
a2
+
b2
,
=
Tan -1
b a
.
Note: If f, g are two functions in x having their nth derivatives then
(f ? g)(n)(x) = f (n) (x) ? g(n) (x) .
Note: If f is a function in x having nth derivative and k R then (kf )(n) (x) = kf (n) (x) .
EXERCISE ? 7 (a)
1. Find the nth derivative of sin3x.
Sol: we know that sin 3x = 3sin x - 4 sin3 x sin3 x = 3sin x - sin 3x
4
Differentiate n times w.r.t x,
( ) dn
dx n
sin3 x
=
1 4
dn dx n
(3sin
x
- sin 3x )
=
1 4
-3n.sin
3x
+
n 2
+
3sin
x
+
n 2
n
z
2. Find the nth derivative of sin 5x. sin 3x.?
Sol: let y = sin 5x.sin 3x = 1 (2sin 5x.sin 3x)
2
y= 1 (cos 2x - cos8x)
2
y = 1 (cos 2x - cos8x)
2
Differentiate n times w.r.t x,
yn
=
1 2
dn dx n
(cos 2x
- cos8x)
yn
=
1 2
2n
cos
2x
+
n 2
-
8n
.
cos
8x
+
n 2
n
z
3. Find nth derivative of ex .cos x.cos 2x
Sol: cos x.cos 2x = 1 (2cos 2x.cos x) = 1 (cos3x + cos x)
2
2
Let y = ex (cos3x + cos x)
2
Differentiate n times w.r.t x,
( ) yn
=
1 2
dn dx n
ex cos 3x + ex cos x
( ) ( ) ( ) ( ) ( ) yn
=
ex 2
10
n
cos
3x + n tan-1 3
n
+
2
n
+ cos
x
+
n
tan-1 1
n
z
= ex 2
n 10
2
cos
3x + n tan-1 3
+
2n
/
2
cos
x
+
n 4
4.
If
y
=
(
x
2
- 1) (
x
-
2)
find
y n
Sol:
Given
y=
2
(x -1)(x - 2)
=
x
1 -
2
-
x
1 -1
(
partial
fractions)
Differentiate n times w.r.t x,
yn
=
2
(-1)n n! ( x - 2)n+1
-
(-1)n n! ( x -1)n+1
=
2
( -1)n
n!
(x
1
) - 2 n+1
-
(
x
1
- 1)n
+1
5.
If
y=
2x +1 ,
x2 - 4
find
y n
Sol:
Let
2x +1 x2 - 4
=
A x-2
+
B x+
2
2x +1 = A(x + 2) + B(x - 2) -----(1)
In (1) ,Put x = 2 5 = A(4) A = 5
4
In (1) , x = -2 -3 = B(-4) B = 3
4
Therefore,
y
=
2x +1 x2 - 4
=
5
4(x -
2)
+
3
4(x +
2)
Differentiate n times w.r.t. x,
yn
=
dn dx n
5
4(x - 2)
+
3
4(x + 2)
yn
=
5 4
(-1)n n! ( x - 2)n+1
+
3 4
(-1)n n! ( x + 2)n+1
=
( -1)n
4
n !
(x
5
) - 2 n+1
+
(x
3
) + 2 n+1
1.
Find the nth derivative of (i)
x
(x -1)2 (x +1)
(ii)
1
(x -1)(x + 2)2
(iii)
x3
(x -1)(x +1)
(iv) x
x2 + x +1
Sol: i)
(v) x +1
x2 - 4
(vi) Log (4x2 - 9)
Let
y=
x
(x -1)2 (x +1)
Resolving into partial fractions
(x
x
-1)2 (x
+ 1)
=
A x -1
+
(x
B
- 1)2
+
C x +1
x = A (x -1)(x +1) + B(x +1) + C(x -1)2 ----- (1
In (1 ), put x = 1 1 = B(1+1) = 2B B = 1
2
In (1 ), put x = -1 -1 = C(-1-1)2 = 4C C = - 1
4
Equating the co . efficient of x2 A + B = 0 A = - 1
2
Therefore,
y
=
-
1
2(x -1)
+
1
2(x -1)2
-
1
4(x +1)
Differentiate n times w.r.t. x,
yn
=
dn dx n
-
2
(
1 x-
1)
+
1
2(x -1)2
-
4(
1 x+
1)
................
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