Chain Rule
Lesson notes
Overview
Differentiate the outside, then multiply by the derivative of the inside. This topic is rated intermediate and is number 62 of 245 in Mathematics. Builds on the basics with relationships, methods and trade-offs.
What you are seeing
The panel above shows a graph of the function. Each slider changes one parameter, so you can see exactly how the curve bends, shifts or stretches. The animation is drawn live from the model, so everything you see responds to the sliders. Press Space to pause, use Restart to replay from the start, and change the speed to slow down quick events.
In depth
The chain rule differentiates a function inside another function. You take the derivative of the outer function, leaving the inner one untouched, and then multiply by the derivative of the inner function. It captures the idea that rates multiply: if a car’s position changes with time and the fuel use changes with position, the fuel use per hour is the product of the two rates. It is probably the most used rule in calculus and is the basis of backpropagation in neural networks.
Reading the relationship
The expression "(f∘g)′ = f′(g(x))·g′(x)" is a compact summary of the idea. It uses the symbols f, g, x; name what each one stands for before you start. Then use the sliders to see how changing one quantity changes the others.
A guided exploration
1. Set every slider to its middle value and note the shape of the curve. 2. Move one slider at a time and describe whether the curve shifts, stretches or turns. 3. Find the value where the behaviour changes sharply, then connect it to the formula.
Try this
Predict the outcome first, then test extreme slider values to find where the pattern changes or breaks.
Common pitfalls
A frequent slip is memorising the rule without the picture. If a result surprises you, sketch the graph or test a simple number by hand before trusting the algebra.
How it connects
It builds on “Product Rule”, the topic before it in Mathematics. Once you are comfortable, continue with “Derivative of eˣ”.
Why it matters
Mathematical ideas like this one are the language that physics, engineering, computing and statistics are written in.
Where you might use it
- Revising “Chain Rule” before a class, exam or certification
- Checking working on homework and exam questions by watching the graph respond
- Modelling growth, change and uncertainty in finance, data science and statistics
- Building the reasoning behind the algorithms and engineering tools people use every day
Frequently asked questions
What is Chain Rule?
Differentiate the outside, then multiply by the derivative of the inside. It is an intermediate topic in Mathematics.
How do I use the Chain Rule simulation?
The panel at the top shows a graph of the function. Each slider changes one parameter, so you can see exactly how the curve bends, shifts or stretches. Move the sliders one at a time, then press Space to pause the animation and look closer.
What does the formula mean?
The expression (f∘g)′ = f′(g(x))·g′(x) packs the idea into symbols. Name each symbol first, then change one quantity with the sliders and see which others respond.
Who is this topic for?
Builds on the basics with relationships, methods and trade-offs. If the quiz feels hard, reread the lesson notes and try the previous topic first.
What should I learn before this?
Start with “Product Rule”, the previous topic in Mathematics. Each topic in a subject builds on the ones before it.
How do I track my progress?
Press “Mark as learned” on this page. Progress is stored only in this browser and is never sent anywhere. The Browse page can show topics you have learned or still need to learn.