Theorem – Chain Rule – Math-reference.net

Derivation of the chain rule.

This proof relies on the definition of a limit.

Theorem

Given a composite function $f(g(x))$ , the derivative of $f(g(x))$ is given by,

$f'(g(x))g'(x)$

Proof

Given an equation $f(x) = ax^2 +bx +c$ , to find the roots of $f$ is equivalent to finding all $x$ such that $ax^2 + bx + c = 0$ . We will also assume that $a ne{0}$ . It follows that,