Question

Suppose that the functions f and g and their derivatives with respect to x have the following values at x = 0 and x = 1: 

x	f(x)	g(x)	f')x)	g'(x)
0	1		5	`(1)/(3)`
1	3	– 4	`-(1)/(3)`	`-(8)/(3)`

(i) The derivative of f[g(x)] w.r.t. x at x = 0 is ......
(ii) The derivative of g[f(x)] w.r.t. x at x = 0 is ......
(iii) The value of `["d"/"dx"[x^(10) + f(x)]^(-2)]_(x = 1_` is ........
(iv) The derivative of f[(x + g(x))] w.r.t. x at x = 0 is ...

Answer 1

(i) `"d"/"dx"{f[g(x)]}`

= `f'[g(x)]."d"/"dx"[g(x)]`
= `f'[g(x)] x g'(x)`

∴ `"d"/"dx"{f[g(x)]}` at x = 0
= `f'[g(0)] x g'(0)`
= `f'(1) x g'(0)`        ...[∵ g(x) = 1 at x = 0]

= `-(1)/(3) xx (1)/(3)`

= `-(1)/(9)`.

(ii) `"d"/"dx"{g[f(x)]}`

= `g'[f(x)]."d"/"dx"[f(x)]`
= `g'[f(x)] x f'(x)`

∴ `"d"/"dx"{f[g(x)]}` at x = 0
= `g'[f(0)] x f'(0)`
= `g'(1) x f'(0)`        ...[∵ f(x) = 1 at x = 0]

= `-(8)/(3) xx 5`

= `-(40)/(3)`.

(iii) `"d"/"dx"[x^10 + f(x)]^-2`

= `2[x^10 + f(x)]^-3."d"/"dx"[x^10 + f(x)]`

= `-2[x^10 + f(x)]^-3 xx [10x^9 + f'(x)]`

∴ `{"d"/"dx"[x^10  f(x)]^-2}_("at" x = 1)`

= `-2[1^10 + f(1)]^-3 xx [10(1)^9 + f'(1)]`

= `(-2)/(1 + 3)^3 xx [10 + (-1/3)]`  ...[∵ f(x) = 3 at x = 1]

= `(-2)/(64) xx (29)/(3)`

= `-(29)/(96)`.

(iv) `"d"/"dx"[f(x + g(x))]`

= `f'(x + g(x))."d"/"dx"[x + g(x)]`
= `f'(x + g(x)) xx [1 + g'(x)]`

∴ `{"d"/"dx"[f(x + g(x))]}_("at" x = 0)`

= `f'(0 + g(0)) x [1 + g'(0)]`
= `f'(1).[1 + g'(0)]`      ...[∵ g(x) = 1 at x = 0]

= `-(1)/(3)[1 + 1/3]`

= `-(1)/(3) xx (4)/(3)`

= `-(4)/(9)`.