Chain Rule
Derivative of a composite function.
Formula
Variables
Example
d/dx sin(x²)=2x cos(x²)
How it works
The chain rule tells you how to differentiate a function built inside another function. If y changes because u changes, and u changes because x changes, then the total rate of change of y with respect to x is the product of those two rates. It comes from the idea that a tiny change in x first affects u, which then affects y, so the effects multiply.
When to use it
Use this whenever you have a composite function, such as a trig, exponential, logarithmic, or power function applied to another expression. It is especially useful in physics, economics, and geometry when one quantity depends on another through an intermediate variable.
Worked example, step by step
Differentiate y = (3x - 1)^4. Let u = 3x - 1, so y = u^4. Differentiate the outside function: dy/du = 4u^3. Differentiate the inside function: du/dx = 3. Apply the chain rule: dy/dx = (dy/du)(du/dx) = 4u^3 · 3. Substitute back u = 3x - 1: dy/dx = 12(3x - 1)^3. Final answer: dy/dx = 12(3x - 1)^3
Common mistakes to avoid
- Differentiating only the outside function and forgetting the derivative of the inside function
- Leaving the answer in terms of u without substituting back to x when required
- Multiplying by the inner derivative in the wrong place or with the wrong sign
Did You Know?
The chain rule is what makes modern neural networks trainable, powering the backpropagation algorithm.
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