Product Rule
Derivative of a product of two functions.
Formula
Variables
Example
d/dx(x·sin x)=sin x + x cos x
How it works
When two functions are multiplied, the derivative is not just the derivative of each one multiplied together because both factors can change at the same time. The product rule captures this by adding two contributions: one from how the first function changes while the second is held fixed, and one from how the second function changes while the first is held fixed. You can think of it as tracking both sources of change in the product.
When to use it
Use this rule whenever you need the derivative of an expression that is explicitly a product of two functions, such as a polynomial times a trig function or a variable coefficient times an exponential. It is common in physics, economics, and engineering whenever one quantity scales another that is also changing with the input.
Worked example, step by step
Differentiate f(x) = x^2 e^x Let u = x^2 and v = e^x Then u' = 2x and v' = e^x Apply the product rule: (uv)' = u'v + uv' f'(x) = (2x)(e^x) + (x^2)(e^x) Simplify: f'(x) = e^x(2x + x^2)
Common mistakes to avoid
- Only differentiating one factor and forgetting the other term
- Multiplying the derivatives, u'v', instead of adding the two product-rule terms
- Treating a product like a sum and applying the power rule to the whole expression
Did You Know?
Leibniz initially got the product rule wrong, mistakenly thinking the derivative of a product was the product of derivatives.
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