The Product and Quotient Rules

The Product Rule

This is another very useful formula:

d (uv) = vdu + udv
dx           dx     dx

This is used when differentiating a product of two functions.

Example

Differentiate x(x² + 1)
let u = x and v = x² + 1
d (uv) = (x² + 1) + x(2x) = x² + 1 + 2x² = 3x² + 1 .
dx

Again, with practise you shouldn't have to write out u = ... and v = ... every time.

The Quotient Rule

d (u/v)  = v(du/dx) - u(dv/dx)
dx                       v²

Example

If y =    x³    , find dy/dx
         x + 4

Let u = x³ and v = (x + 4). Using the quotient rule, dy/dx =
(x + 4)(3x²) - x³(1)  =   2x³ + 12x²
        (x + 4)²                 (x + 4)²