Product Rule

If y = uv , where u and v are functions of x,

then

\( \normalsize \frac{dy}{dx} = \scriptsize u \normalsize \frac{dv}{dx} + \scriptsize v \normalsize \frac{du}{dx}\)

Example: Differentiate \( \scriptsize (3x -2)(x^2 + 3) \) with respect to x

let u = (3x – 2) and v = x² + 3

\( \frac{du}{dx} \scriptsize = 3\)

\( \frac{dv}{dx} \scriptsize = 2x \)

\( \normalsize \frac{dy}{dx} = \scriptsize u \normalsize \frac{dv}{dx} \times \scriptsize v \normalsize \frac{du}{dx}\)

\( \frac{dy}{dx} = \scriptsize (3x \; – \; 2) (2x) + (x^2 + 3) (3)\)

\( \frac{dy}{dx} = \scriptsize 6x^2 -4x + 3x^2 + 9 \)

=\( \frac{dy}{dx} = \scriptsize 9x^2 – 4x + 9 \)