The Product Rule Calculator differentiates expressions formed by multiplying two functions. It is useful when both factors depend on the variable and neither can simply be treated as a constant.
Product Rule Formula
For y = f(x)g(x), the derivative is y′ = f′(x)g(x) + f(x)g′(x). Each factor is differentiated once while the other factor remains unchanged in that term.
How to Work With the Tool
Enter the two factors in the fields provided by the calculator. Keep each factor grouped correctly, especially when it contains a sum, power, or trigonometric expression. After calculating, inspect whether both product-rule terms appear in the result.
Example
Take y = x² sin(x). Let f=x² and g=sin(x). Then f′=2x and g′=cos(x). The derivative is 2x sin(x) + x² cos(x).
Why the Rule Is Needed
Multiplying the two functions first may make the expression longer or impossible to expand neatly. The product rule preserves the original structure and works equally well when the factors are polynomials, trigonometric functions, exponentials, or combinations of them.
Common Error
Do not differentiate both factors in the same term. The correct pattern contains two terms: one differentiates the first factor and the other differentiates the second. Comparing the calculator result with this pattern is a simple accuracy check.