unit 1 differential calculus - IGNOU

(iii) (x 2 + x + 6) sin x
(iv) (x + 2) (x 2 + 1) + cos x
(b) Find the derivatives of the functions
(i)
2x
+ 3
x
2 −
(ii) tan x
(iii)
( 1
1 +
5
− sin x)
2
cos
x
(c) Find the 2 nd order derivatives of the functions
(i) sin 2 x
(ii) ax 3 + bx 2 + cx + d
1.6.6 Derivatives of Some Elementary Functions
Power Function y = x α (α is real)
We have already proved that the power function y = x n , where n is a
positive integer, has the derivative
dy
dx
=
n
n x
−1
We shall show now that the above rule also holds when
integers with q > 0).
Let us assume p, q > 0.
Then y q = x p . By the method of implicit differentiation, we get
or
q y
dy
= p x
dx
q −1
p −1
p −1
p −1
p
−1
dy p x p x p q n
= = = x = nx
dx q −1
p
q y
p − q
q
q x
−1
p
q
y = x (q, p are
,
if
p
q
= n
Differential Calculus
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