Product rule

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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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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:

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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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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