D - Chain Rule Lesson

Chain Rule

Derivative of a Composite Function

When given a composite function F(x) = f(g(x)) or alternatively F = f ο g, for which derivatives of both f and g exist, the derivative of the composite function f ο g is the product of the derivatives of f and g. This fact is one of the most important of the differentiation rules and is known as the Chain Rule. Note that when using the Chain Rule, we work from the outside to the inside, i. e., we differentiate the outer function f at the inner function g(x) and then multiply by the derivative of the inner function.

View the presentation below on the Chain Rule derivation.

General Power Rule

A special case of the Chain Rule where the outer function f is a power function is noteworthy. If n is any real number and y = [g(x)]ⁿ, then we can write

y = f(u) = uⁿ where u = g(x). By using the Chain Rule and then the Power Rule, the general Power Rule is obtained: $y'=f'\left(u\right)=\frac{du^n}{dx}=nu^{n-1}\cdot dx$.

View examples of how to use the Chain Rule to find higher order derivatives.

View the presentation on the Doppler effect, an application of the Chain Rule.

Trigonometric Functions and the Chain Rule

All formulas for differentiating trigonometric functions can be combined with the Chain Rule as the information below illustrates:$\frac{d}{dx}\sin u=\cos u\frac{du}{dx}\\ \frac{d}{dx}\cos u=-\sin u\frac{du}{dx}\\ \frac{d}{dx}\tan u=\sec^{2} u\frac{du}{dx}\\ \frac{d}{dx}\cot u=-\csc^{2} u\frac{du}{dx}\\ \frac{d}{dx}\sec u=(\sec u)(\tan u)\frac{du}{dx}\\ \frac{d}{dx}\csc u=-(\csc u)(\cot u)\frac{du}{dx}\\$

View the presentation illustrating the Chain Rule with trigonometric functions from the beginning to 7:10.

Chain Rule Practice

Chain Rule: Even More Problems!

Complete problems from your textbook and/or online resources as needed to ensure your complete understanding of the Chain Rule.

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