Chain Rule Formula

The Chain Rule is a formula for computing the derivative of the composition of two or more functions. For instance, if f and g are functions, then the chain rule expresses the derivative of their composition.
d/dx [f(g(x))] = f'(g(x)) g'(x)

The Chain Rule Formula is as follows – 

\[\LARGE \frac{dy}{dx}=\frac{dy}{du}.\frac{du}{dx}\]

Solved Examples

Example 1: Differentiate y = cos $x^{2}$



y = cos $x^{2}$

Let u = $x^{2}$, so that y = cos u

Therefore: $\frac{du}{dx}$=2x

$\frac{dy}{du}$ = -sin u

And so, the chain rule says:


$\frac{dy}{dx}$= -sin u $\times$ 2x

= -2x sin $x^{2}$

Example 2: 

Differentiate f(x) = (1 + x2)5.


Using the Chain rule,

dy/dx = dy/du ⋅ du/dx

Let us take y = u5 and u = 1 + x2

Then dy/du = d/du (u5) = 5u4

du/dx = d/dx (1 + x2 )= 2x

dy/dx = 5u4⋅2x = 5(1 + x2)4⋅2x

= 10x(1 + x2)4

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