Chain Rule Formula

The Chain Rule formula is a formula for computing the derivative of the composition of two or more functions. Chain rule in differentiation is defined for composite 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)

What is Chain Rule Formula?

The Chain Rule Formula is as follows – 

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

Solved Examples on Chain Rule Formula

Example 1: Differentiate y = cos x2

Solution:

Given,

y = cos x2

Let u = x2, so that y = cos u

Therefore;

\(\begin{array}{l}\frac{du}{dx}=2x\end{array} \)

\(\begin{array}{l}\frac{dy}{du} = -sin u\end{array} \)

And so, the chain rule says:

\(\begin{array}{l}\frac{dy}{dx}=\frac{dy}{du}.\frac{du}{dx}\end{array} \)

\(\begin{array}{l}\frac{dy}{dx}= -sin u \times 2x\end{array} \)

= -2x sin x2

Example 2: 

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

Solution: 

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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