Calculus – Chain Rule

The Chain Rule is used to find the derivative of composite functions.

The Chain Rule

If f and g are both differentiable and F(x) is the composite function defined by F(x) = f(g(x)) then F is differentiable and F ’ is given by the product

F ’(x) = f ‘(g(x)) g’(x)

In Leibniz notation, if y = f(u) and u = g(x) are both differentiable functions, then

Note: In the Chain Rule, we work from the outside to the inside. We differentiate the outer function and then we multiply with the derivative of the inner function.

Example:

Find the derivatives of each of the following

Solution:

Example:

Differentiate y = (2x + 1)⁵(x³ – x +1)⁴

Solution:

In this example, we use the Product Rule before using the Chain Rule.

Videos

An introduction to the chain rule - In this video, Professor Edward Burger introduces the chain rule.

Using the chain rule - In this video, Professor Edward Burger teaches how to use the chain rule.

Custom Search