Quotient rule

<math>\frac{d}{dx} \sin x=?\,</math>

This article/section deals with mathematical concepts appropriate for late high school or early college.

The Quotient Rule is a rule in calculus pertaining to the derivative of a variable or function divided by another. It can be written as follows:

<math>\frac{d}{dx} \left(\frac{u}{v}\right) = \frac{v \left(\frac{du}{dx}\right) - u \left(\frac{dv}{dx}\right)}{v^2}</math>

Alternatively, in prime notation, it can be written as:

<math>\left(\frac{u}{v}\right)' = \frac{u'v - v'u}{v^2}</math>

It is easily remembered by the rhyme "Low 'D' High minus High 'D' Low, draw the line, and square below" (the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator all over the denominator squared).

Example

Let f(x) be the function:

<math> f(x) = \frac{\sin{3x}}{x^2} </math>

and suppose we want to find its derivative, f'(x). By applying the quotient rule we get:

<math> f'(x) = \frac{x^2 \times (3 \cos{3x}) - \sin{3x} \times (2x)}{(x^2)^2} </math>

and so the derivative of f(x) is simply:

<math> f'(x) = \frac{3x \cos{3x} - 2\sin{3x}}{x^3} </math>

Rules for finding derivatives

• Power rule
• Constant-multiple rule
• sum rule
• Chain rule
• Product rule
• Quotient rule