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Chain Rule Calculator

Chain-rule derivative calculator. Enter inner u(x) and outer f(u) to compute y′=f′(u)·u′(x) for y=f(u(x)), step by step—great for chain-rule practice.

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y = f(u(x)) ⇒ dy/dx = f′(u)·u′(x)
Use x
Use the variable u
Compute the composite derivative here

📖 Tutorial｜Chain Rule

1. Definition
If y=f(u) and u=u(x), then dy/dx = f′(u)·u′(x): derivative of the outer times derivative of the inner.
2. Symbols
SymbolMeaning
u(x)Inner function
f(u)Outer function
u′(x)Derivative of the inner
f′(u)Derivative of outer w.r.t. u
3. How it works
  • We numerically compute u′(x) and f′(u);
  • Their product is the composite derivative;
  • The outer expression must use the variable u.
4. Steps
  1. Enter inner u(x) using x;
  2. Enter outer f(u) using u;
  3. Enter x and click Calculate.
5. Example
Example: y=sin(x²). u=x², f(u)=sin u. u′=2x=4, f′(u)=cos u=cos4, so y′=4cos4≈−2.615.
6. Pitfalls
The outer function must be written in u, not x;
Never forget to multiply by u′;
For nested composites, multiply through every layer.

❓ FAQ｜Chain Rule

What is the chain rule mantra?
“Outside in, differentiate layer by layer, multiply.” Differentiate the outer function, then multiply by the inner derivative.
Why write the outer in u?
So the tool can differentiate each layer independently and multiply via the chain rule.
How to differentiate a triple composite?
For f(g(h(x))), the derivative is f′·g′·h′.
Chain rule vs product rule?
Chain handles nested f(g(x)); product handles side-by-side f(x)·g(x).
Why the small numerical error?
Both u′ and f′(u) use central differences, error around 1e-5.