Chain Rule

Unit 2 · Differentiation — Rules

What is the Chain Rule?

The Chain Rule tells you how to differentiate a composite function — a function inside another function. If y = f(g(x)), you multiply the derivative of the outer function (evaluated at the inner) by the derivative of the inner function.

Chain Rule Formula

Leibniz notation

\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}

Prime notation

\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)

Common Patterns

Power

\frac{d}{dx}[u^n] = n u^{n-1} \cdot u'

Exponential

\frac{d}{dx}[e^u] = e^u \cdot u'

Sin / Cos / Tan

\frac{d}{dx}[\sin u] = \cos u \cdot u', \quad \frac{d}{dx}[\cos u] = -\sin u \cdot u'

Natural log

\frac{d}{dx}[\ln u] = \frac{u'}{u}

Type 1

Basic Composition

A function nested inside another — power, trig, exponential, or natural log. One clean application of the chain rule.

Example 1

Find the derivative of the function.

f(x) = (3x^2 + 1)^5

Example 2

Find the derivative of the function.

g(x) = \sin(x^3)

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Type 2

Chain Rule + Product Rule

When the function is a product of two expressions and at least one requires the chain rule. Apply the product rule first, then chain rule inside.

Example 3

Find the derivative of the function.

h(x) = x^2 e^{\sin x}

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Type 3

Evaluate at a Point

Find the derivative using the chain rule, then substitute a specific x value to get a numerical answer.

Example 4

Find f′(0) for the function.

f(x) = \ln(x^2 + e)

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