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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 ...
Solution
(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)`.
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