Chain Rule
Derivative of a composite function.
Key idea
Derivative of a composite function.
In depth
(f(g(x)))′ = f′(g(x)) · g′(x). For nested functions, differentiate the outer function, leaving the inside alone, and multiply by the derivative of the inside. It reflects how rates multiply.
Worked example
d/dx sin(x²) = cos(x²) · 2x. d/dx (3x + 1)⁵ = 5(3x + 1)⁴ · 3.
Formula
Real-world uses
- Computing rates in linked systems such as a balloon’s radius changing with its volume
- Neural network training
Common mistake
Forgetting the inner derivative.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Chain Rule?
Derivative of a composite function. It is a topic at the advanced level in Calculus.
What is the formula for Chain Rule?
The key formula is (f(g(x)))′ = f′(g(x))·g′(x).
Where is Chain Rule used in real life?
Real-world examples: Computing rates in linked systems such as a balloon’s radius changing with its volume; Neural network training.
What should I learn before Chain Rule?
Make sure you are comfortable with Quotient Rule, the previous topic in Calculus.
What is a common mistake with Chain Rule?
Forgetting the inner derivative.
How can I practise Chain Rule?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Chain Rule?
Move on to Derivatives of Trig Functions next.
Related advanced topics in Calculus
- Idea of a Limit – The value a function approaches as the input approaches a point.
- One-Sided Limits – Left and right limits must agree for the limit to exist.
- Limit Laws – Limits of sums, products and quotients are computed from the parts.
- Factoring and Rationalizing Limits – Remove 0/0 indeterminacy with algebra.
- Limits at Infinity – End behaviour of functions.
- Infinite Limits and Asymptotes – Function values grow without bound near a vertical asymptote.