The Chain Rule

Definition

The Chain Rule is the formula for calculating the derivative of a composite function $f(g(x))$, relating the rate of change of the outer function to the rate of change of the inner function.

Why It Matters

It enables the differentiation of complex, composite functions, making it possible to model how changes in one variable propagate through an entire chain of dependencies.

Core Concepts

• Formal Theorem: $(f \circ g)'(x) = f'(g(x)) \cdot g'(x)$.
  • How to read: “The quantity f circle g prime of x equals f prime of g of x times g prime of x.”
  • Meaning / when to use: Differentiate the outer function $f$ while keeping the inner $g(x)$ intact, then multiply by the derivative of the inner. Essential for nested functions like $\sin(x^2)$ or $e^{3x}$.
• Leibniz Form: $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$.
  • How to read: “The derivative d y d x equals the derivative d y d u times the derivative d u d x.”
  • Meaning: If $y$ depends on $u$ and $u$ depends on $x$, the rates multiply along the chain—like canceling $du$ in a fraction (formally justified by differentials).
• Operation: Differentiate the “outside” function (keeping the “inside” as a single unit), then multiply by the derivative of the “inside.”

Connected Concepts

• Chain Rule for Multivariable Functions
• Determinants and Cramer’s Rule
• The LIATE Rule