Product rule

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

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

The Product Rule is a rule in calculus pertaining to the derivative of two variables or functions. The Leibniz notation is as follows:

<math>\frac{d}{dx} (uv) = u \frac{dv}{dx} + v \frac{du}{dx}</math>

The product rule in words is the derivative of uv is u multiplied by the derivative of v plus the derivative of u multiplied by v. The commutative property allows the terms to be interchanged, which occasionally leads to introduction in reverse.

Example

Let f(x) be the function:

<math> f(x) = 3x \sin{5x} </math>

To find its derivative, we split f(x) in two and apply the product rule with u equal 3x and v equal to the sine term:

<math> f'(x) = 3 \times \sin{5x} + 3x \times (5\cos{5x}) = 3\sin{5x} + 15x \cos{5x} </math>

Rules for finding derivatives

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