Theorem – Product Rule – Math-reference.net

Derivation of the product rule.

This proof relies on the definition of the derivative for a single variable function at $c$ .

Theorem

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

$displaystyle{f'(g(x))g'(x)}$

Proof

Given an equation $displaystyle{f(x) = ax^2 +bx +c}$ , 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,