Question

Question:-
Explain the Chain rule or Function of a function rule. Give one or two examples.
[ Explanation needed.]​

Answer 1

Chain rule / Function of function rule :

Let y = f(u) , where y is a function of u and u is a function of x .

Then , $\sf \dfrac{dy}{dx} = \dfrac{dy}{du} \times \dfrac{du}{dx}$

Examples :

$\bullet\ \; \sf \dfrac{ d }{ dx } (x^2) = 2x$

Here , y = f(x) = x²

In the same way ,

$\bullet\ \; \sf \dfrac{ d }{ dx } \left[ (sin\ x)^2 \right] = 2\ sin\ x\ cos\ x = sin\ 2x$

Here , y = f(u) = (u)²

Again , u is a function of x i.e., u = sin x

$\implies \sf \dfrac{dy}{dx}$

$\implies \sf \dfrac{dy}{du} \times \dfrac{du}{dx}$

$\implies \sf \dfrac{d }{ du } (u)^2 \times \dfrac{ d }{ dx } (sin\ x)$

$\implies \sf 2\ u\ cos\ x$

$\implies \sf 2\ sin\ x\ cos\ x$

$\implies \sf sin\ 2x$

$\bullet\ \; \sf \dfrac{ d }{dx} ( \log x)^3 = \dfrac{3( \log x)^2 }{x}$

Here , y = u³ & u = ㏒ x

$\implies \sf \dfrac{dy}{dx}$

$\implies \sf \dfrac{dy}{du} \times \dfrac{du}{dx}$

$\implies \sf \dfrac{d}{du}(u^3) \times \dfrac{d}{dx} ( \log x)$

$\implies \sf 3\ u^2 . \dfrac{1}{x}$

$\implies \sf \dfrac{3 ( \log x)^2}{x}$

Answer 2

chain rule: is a formula for computing the derivative of the composition of two or more functions.

if f and g are the functions, then the derivative of the composite function fog in terms of the derivatives of f and g.

for example , the chain rule for

$\frac{d}{dx} (fog)(x) = \frac{df}{dg} . \frac{dg}{dx}$

for example:

$\frac{d}{dx}( {sin \: x}^{2} ) = (\frac{d}{dx²}sin \: x \: ²).(\frac{d}{dx}x²) \\ = cos \: x \: ² \: . \: 2x \\ = 2xcos \: x \: ²$