The derivative of a composite function f(g(x)) equals f'(g(x)) \cdot g'(x): the derivative of the outer function evaluated at the inner, times the derivative of the inner. The chain rule is essential for differentiating almost all real-world functions, which are composites.
This concept is covered in depth in our power rule and chain rule explained, with worked examples, practice problems, and common mistakes.
Definition
The derivative of a composite function f(g(x)) equals f'(g(x)) \cdot g'(x): the derivative of the outer function evaluated at the inner, times the derivative of the inner.
💡 Intuition
Derivative of outside times derivative of inside. Unpack layers.
🎯 Core Idea
To differentiate a composite function, multiply the derivative of the outer by the derivative of the inner.
Example
Formula
Notation
In Leibniz notation: \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} where y = f(u) and u = g(x).
🌟 Why It Matters
The chain rule is essential for differentiating almost all real-world functions, which are composites.
💭 Hint When Stuck
Write u = [inner function] on scratch paper, then differentiate the outer function with respect to u, and multiply by du/dx.
Formal View
Related Concepts
🚧 Common Stuck Point
Identify 'inside' and 'outside' functions first, then apply.
⚠️ Common Mistakes
- Forgetting to multiply by the derivative of the inner function: \frac{d}{dx}[\sin(x^2)] = \cos(x^2) \cdot 2x, not just \cos(x^2).
- Applying the chain rule only once when there are multiple layers of nesting: for \sin(e^{3x}), you need \cos(e^{3x}) \cdot e^{3x} \cdot 3 — three layers, three factors.
- Confusing when to use the chain rule vs. the product rule: \sin(x) \cdot x^2 needs the product rule, while \sin(x^2) needs the chain rule.
Go Deeper
Frequently Asked Questions
What is Chain Rule in Math?
The derivative of a composite function f(g(x)) equals f'(g(x)) \cdot g'(x): the derivative of the outer function evaluated at the inner, times the derivative of the inner.
Why is Chain Rule important?
The chain rule is essential for differentiating almost all real-world functions, which are composites.
What do students usually get wrong about Chain Rule?
Identify 'inside' and 'outside' functions first, then apply.
What should I learn before Chain Rule?
Before studying Chain Rule, you should understand: derivative, composition.
Prerequisites
Cross-Subject Connections
How Chain Rule Connects to Other Ideas
To understand chain rule, you should first be comfortable with derivative and composition.
Want the Full Guide?
This concept is explained step by step in our complete guide:
Derivatives Explained: Rules, Interpretation, and Applications →