Chain Rule
Differentiate the outside function then multiply by the derivative of the inside.
About this topic
The chain rule differentiates composite functions: if y = f(g(x)), then y′ = f′(g(x)) · g′(x). Differentiate the outer function, leaving the inside alone, then multiply by the derivative of the inside. For (3x + 1)⁵ this gives 5(3x + 1)⁴ · 3.
In Leibniz notation, dy/dx = dy/du · du/dx, which looks like cancelling fractions and reflects chains of rates. If a car uses fuel at a rate depending on speed, and speed changes with time, the rate of fuel use over time is the product of the two rates. For several nested functions, work from the outside in and multiply each layer.
Example: for y = sin(x²), the outer function is sine and the inner is x², so y′ = cos(x²) · 2x. For y = √(1 + x³), y′ = (1/(2√(1 + x³))) · 3x² = 3x²/(2√(1 + x³)).
The chain rule is used throughout calculus, in related rates, implicit differentiation and in training neural networks, where it is called backpropagation. Common mistakes are forgetting the inner derivative and differentiating the inside function when the outside should be treated first. Tip: identify the layers by asking which operation you would do last when evaluating the function.
Key ideas
- d/dx f(g(x)) = f′(g(x))·g′(x)
- Peel the layers
- Essential for composite functions
Quick check
Use cases: where Chain Rule helps
- Homework: review the key idea behind Chain Rule before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Chain Rule without notes.
- Visual learning: play with the interactive lab to see how Chain Rule changes when you adjust its inputs.
- Teaching a friend: explain Chain Rule in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Chain Rule?
Differentiate the outside function then multiply by the derivative of the inside.
What will I learn in Chain Rule?
d/dx f(g(x)) = f′(g(x))·g′(x). Peel the layers. Essential for composite functions.
Is Chain Rule part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Chain Rule lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Chain Rule?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.