Chain rule

Definition

The chain rule is a formula for computing the derivative of the composition of two or more functions. It states that if $y = f(g(x))$, then the derivative $dy/dx = f'(g(x)) * g'(x)$.

5 Must Know Facts For Your Next Test

1. The chain rule is essential for differentiating composite functions.
2. It can be extended to compositions of more than two functions.
3. The inner function is differentiated first, followed by the outer function.
4. The chain rule applies to implicit differentiation as well.
5. The notation $(f \circ g)'(x) = f'(g(x)) * g'(x)$ is often used to represent the chain rule.

Review Questions

• What is the derivative of $h(x) = \sin(3x^2)$ using the chain rule?
• Explain how to apply the chain rule to differentiate $y = (2x + 1)^5$.
• Differentiate $z = e^{\cos(x)}$ using the chain rule.

Related terms

Derivative: A measure of how a function changes as its input changes; it represents an instantaneous rate of change.

Composite Function: A function formed when one function is substituted into another, such as $f(g(x))$.

Implicit Differentiation: A method used to find derivatives when a function is not given in explicit form, often involving the use of the chain rule.