Chain Rule Formula

Q: What is the Chain Rule formula?

The derivative of a composite function $f(g(x))$ equals $f'(g(x)) \cdot g'(x)$: the derivative of the outer function evaluated at the inner, times the derivative of the inner.

Q: What do the symbols mean in the Chain Rule formula?

In Leibniz notation: $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$ where $y = f(u)$ and $u = g(x)$.

The Formula

(f \circ g)'(x) = f'(g(x)) \cdot g'(x)

When to use: Derivative of outside times derivative of inside. Unpack layers.

Quick Example

\frac{d}{dx}(\sin(x^2)) = \cos(x^2) \cdot 2x Outside: \sin. Inside: x^2.

Notation

In Leibniz notation: \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} where y = f(u) and u = g(x).

What This Formula Means

The derivative of a composite function f(g(x)) equals f'(g(x)) \cdot g'(x): the derivative of the outer function evaluated at the inner, times the derivative of the inner.

Derivative of outside times derivative of inside. Unpack layers.

Formal View

If g is differentiable at x and f is differentiable at g(x), then (f \circ g)'(x) = f'(g(x)) \cdot g'(x).

Worked Examples

Example 1

easy

Find the derivative of f(x) = (3x + 1)^4.

Solution

1
Identify the outer function u^4 and the inner function u = 3x + 1.
2
Apply the chain rule: \frac{d}{dx}[u^4] = 4u^3 \cdot \frac{du}{dx}.
3
The derivative of the inner function: \frac{du}{dx} = 3.
4
Combine: f'(x) = 4(3x+1)^3 \cdot 3 = 12(3x+1)^3.

Answer

f'(x) = 12(3x + 1)^3

The chain rule says: differentiate the outer function, keep the inner function, then multiply by the derivative of the inner function. Think of it as peeling layers.

Example 2

medium

Find the derivative of f(x) = (x^2 + 1)^5.

Example 3

hard

Find the derivative of f(x) = \sin(x^3).

Common Mistakes

Forgetting to multiply by the derivative of the inner function: \frac{d}{dx}[\sin(x^2)] = \cos(x^2) \cdot 2x, not just \cos(x^2).
Applying the chain rule only once when there are multiple layers of nesting: for \sin(e^{3x}), you need \cos(e^{3x}) \cdot e^{3x} \cdot 3 — three layers, three factors.
Confusing when to use the chain rule vs. the product rule: \sin(x) \cdot x^2 needs the product rule, while \sin(x^2) needs the chain rule.

Why This Formula Matters

The chain rule is essential for differentiating almost all real-world functions, which are composites.

Frequently Asked Questions

What is the Chain Rule formula?

The derivative of a composite function f(g(x)) equals f'(g(x)) \cdot g'(x): the derivative of the outer function evaluated at the inner, times the derivative of the inner.