Product, quotient, and chain rules

The power rule handles sums of powers. But functions get built by multiplying, dividing, and nesting simpler ones — and each of those needs its own rule.

Product and quotient

You can't just multiply derivatives. The correct rules:

For the product, differentiate one factor at a time and add. Example: for f(x) = x²·sin x, take u = x² (u′ = 2x) and v = sin x (v′ = cos x):

f′(x) = 2x·sin x + x²·cos x.

The chain rule

The one you'll use most is for composed functions — a function inside a function. Its rate is the product of the rates:

If y = f(g(x)), then dy/dx = f′(g(x)) · g′(x). Differentiate the outer function (leaving the inner alone), then multiply by the derivative of the inner. Leibniz notation makes it look like fractions cancelling: dy/dx = (dy/du)(du/dx).

Peeling the layers

For y = (x² + 1)³, identify the layers and work outside-in:

• Outer: something cubed → derivative 3·(something)²
• Inner: x² + 1 → derivative 2x

Multiply: dy/dx = 3(x² + 1)² · 2x = 6x(x² + 1)². The outer rule leaves the inner untouched; the "· 2x" is the chain rule paying the inner its due.

Why this matters

Real models are layered — a temperature that depends on time, feeding a rate that depends on temperature. The chain rule is how those rates combine, and it's the single most-used differentiation rule in all of applied calculus.