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
- The chain rule is essential for differentiating composite functions.
- It can be extended to compositions of more than two functions.
- The inner function is differentiated first, followed by the outer function.
- The chain rule applies to implicit differentiation as well.
- 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.
"Chain rule" appears in:
Subjects (1)
Study guides (1)
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.
