Chain rule

f

is differentiable at

a

and

g

is differentiable at

f (a)

, then

(g \circ f)^{'} (a) = g^{'} (f (a)) f^{'} (a) .

(1)

Table of contents
- Nondifferentiable inner function with a differentiable nondegenerate composite Chain rule
- Second derivative chain rule Chain rule

Nondifferentiable inner function with a differentiable nondegenerate composite

 0  0

At zero,

f (x) = 3 x

is not differentiable and

g (y) = y^{3}

is differentiable, while

g \circ f (x) = x

has derivative one.

Second derivative chain rule

 0  0

For twice differentiable functions,

(g \circ f)^{''} (a) = g^{''} (f (a)) [f^{'} (a)]^{2} + g^{'} (f (a)) f^{''} (a) .

(1)

 Ancestors (7)

 Incoming links (231)

 View article source

 Discussion (0)

New discussion

There are no discussions about this article yet.

 Articles by others on the same topic (2)

Chain rule by

Donald Trump  29 Created 2023-01-15 Updated 2025-04-18

 Read the full article

Chain rule by

Ciro Santilli  40 Updated 2025-07-16

 View more

Here's an example of the chain rule. Suppose we want to calculate:

\frac{d}{d x} e^{2 x}

(1)

So we have:

f (x) = e^{x} g (x) = 2 x

(2)

and so:

f^{'} (x) = e^{x} g^{'} (x) = 2

(3)

Therefore the final result is:

f^{'} (g (x)) g^{'} (x) = e^{2 x} 2 = 2 e^{2 x}

(4)

 Read the full article

Chain rule by

Barack Obama  11 Created 2023-01-13 Updated 2025-04-18

 Read the full article

Chain rule by

Wikipedia Bot  1

 View more

The chain rule is a fundamental concept in calculus used to differentiate composite functions.

 Read the full article

  See all articles in the same topic Create my own version

Chain rule

Nondifferentiable inner function with a differentiable nondegenerate composite

Second derivative chain rule

 Ancestors (7)

 Incoming links (231)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (2)

 Discussion (0)