Chain Rule in Differentiation

Chain Rule in Differentiation

How to solve the chain rule in differentiation problems: formula, proof, examples, and their solutions.

Formula

[f(g(x))]' = f'(g(x))*g'(x)

The chain rule is a way
to differentiate a composite functioncomposite function: f(g(x)).

[f(g(x))]' = f'(g(x))⋅g(x)

First differentiate the outer function
without touching the inner function:
f'(g(x)).

Then differentiate the inner function:
g'(x).

Proof

Chain Rule in Differentiation: Proof of the Formula

Start from y = f(g(x)).

Set t = g(x).

Then dt/dx = g'(x).

Put t = g(x) into y = f(g(x)).

Then y = f(t).

Then dy/dt = f'(x).

y = f(g(x))

So [f(g(x))]' = dy/dx
= (dy/dt)⋅(dy/dx).

Put f'(t) and g'(x)
into (dy/dt) and (dt/dx).

Put g(x) into the t.

Then [f(g(x))]' = f'(g(x))⋅g(x).

Example 1

Find the derivative of the given function. y = (2x^2 - 1)^8

y = (2x2 - 1)8

y' is equal to,
the derivative of the outer part, 8(2x2 - 1)7
times, the derivative of the inner part, (2⋅2x1 - 0).

Derivatives of polynomials

Example 2

Find the derivative of the given function. y = 1/(x^3 + 2x)

Previously, you've solved this example.

Let's solve the same example
by using the chain rule.

Change 1/(x3 + 2x) into (x3 + 2x)-1.

y = (x3 + 2x)-1

y' is equal to,
the derivative of the outer part, (-1)⋅(x3 + 2x)-2
times, the derivative of the inner part, (3⋅2x1 + 2⋅1x0).

Derivatives of polynomials

As you can see,
you got the same answer.

So, to differentiate a function in reciprocal form,
you can either use the reciprocal rule
or use the chain rule.