Chain Rule Calculator
Calculate derivatives of composite functions using d/dx[f(g(x))] = f’(g(x))·g’(x). Find derivatives of nested functions step by step.
∫ Calculus📐 d/dx[f(g(x))] = f’(g(x)) × g’(x)🔢 Math
Outer function
Outer exponent n
Inner coefficient a
Inner exponent m
Please enter valid values.
Formula & Reference
| Variable | Symbol | Formula | Units |
|---|---|---|---|
| Chain Rule Calculator | — | d/dx[f(g(x))] = f’(g(x)) × g’(x) | expression |
Step-by-Step Examples
Example 1
(2x)³
d/dx[(2x)³]
- outer: 3u², inner: 2x→2
- 3(2x)²·2=24x²
✓ 24x²
Example 2
sin(x²)
d/dx[sin(x²)]
- outer: cos(u), inner: x²→2x
- cos(x²)·2x
✓ 2x·cos(x²)
Example 3
e^(3x)
d/dx[e^(3x)]
- outer: e^u, inner: 3x→3
- e^(3x)·3
✓ 3e^(3x)
Real-World Applications
ML
Backpropagation chains derivatives through layers.
Physics
Composite motion equations.
Engineering
Heat transfer and signal processing.
Economics
Nested functions in compound growth.
Common Mistakes to Avoid
⚠️
Missing inner derivative
Must multiply outer AND inner derivatives.
⚠️
Wrong identification
Outermost operation is the outer function.
⚠️
Not seeing composites
sin(x²) and (3x+1)⁵ both need chain rule.
Frequently Asked Questions
When to use chain rule? ▾
When you have f(g(x)): sin(x²), e^(3x), (2x+1)⁵.
Leibniz notation? ▾
dy/dx = (dy/du)·(du/dx).
Triple composite? ▾
Multiply three derivatives: f'(g(h))·g'(h)·h'(x).
Generalized power rule? ▾
d/dx[(g)^n] = n·(g)^(n-1)·g'(x).
e^(5x)? ▾
Outer: e^u→e^u. Inner: 5x→5. Result: 5e^(5x).
Related Math Calculators
Formula Explorer connections
Interpretation: differentiate the outer function while leaving the inner expression in place, then multiply by the inner derivative. Assumption: both functions must be differentiable at the relevant points; domain restrictions of the composition still apply.