Product rule

${\frac {d}{dx}}\sin x=?\,$

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:

${\frac {d}{dx}}(uv)=u{\frac {dv}{dx}}+v{\frac {du}{dx}}$

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:

$f(x)=3x\sin {5x}$

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:

$f'(x)=3\times \sin {5x}+3x\times (5\cos {5x})=3\sin {5x}+15x\cos {5x}$

Rules for finding derivatives

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