Product Rule

Definition

The product rule is a formula used to find the derivative of two functions multiplied together.

Given two differentiable functions f(x) and g(x), the derivative with respect to x of the product f(x)g(x) is given by:

ddx[f(x)g(x)]=f(x)g(x)+f(x)g(x),

(A prime denotes the derivative with respect to x, i.e. f(x)=dfdx.)

Note: the product rule can be extended to compute the derivative of a product of more than two functions.

Example: Given three functions f(x), g(x) and h(x), the derivative with respect to x of the product f(x)g(x)h(x) is:

ddx[f(x)g(x)h(x)]=f(x)g(x)h(x)+f(x)(g(x)h(x))=f(x)g(x)h(x)+f(x)g(x)h(x)+f(x)g(x)h(x)

Worked Example

Example 1

Given f(x)=4x3sinhx, find dfdx.

Solution

Let g(x)=4x3 and h(x)=sinh(x). Then by the product rule:

dfdx=g(x)h(x)+g(x)h(x)=(ddx[4x3])sinhx+4x3(ddx[sinhx])=12x2sinhx+4x3coshx

Video Examples

Example 1

Prof. Robin Johnson differentiates x2e3x.

Example 2

Prof. Robin Johnson differentiates (3x2)3(2x+1)4.

Test Yourself

Test yourself: Numbas test on the product rule

Test yourself: Numbas test on differentiation, including the chain, product and quotient rules

See Also

External Resources

Whiteboard maths

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