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 (3x−2)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
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