The idea

The chain rule

Differentiating a function of a function — y = f(g(x)) — by multiplying the outside derivative by the inside derivative, with the dy/dx = (dy/du)(du/dx) substitution method and the reciprocal connection dx/dy = 1 / (dy/dx).

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Pure · Differentiation

The chain rule

Why it works

Nested functions

The power rule handles

x^n

, but real functions are usually nested: a curve like

y = (3x + 1)^5

is one function — "raise to the fifth" — wrapped around another — "triple and add one". You could expand the bracket and differentiate term by term, but for

(3x+1)^5

that is grim, and for

y = \sqrt{x^2 + 1}

it is impossible. The chain rule differentiates the nesting directly.

Name the inside

Give the inside its own name. Write

u = 3x + 1

, so the outer function is

y = u^5

. Now you have two easy derivatives, and the chain rule says multiply them:

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

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