Product Rule (differentiation)
The product rule states that the derivative of the product of two differentiable functions equals the first function times the derivative of the second, plus the second function times the derivative of the first. It is a fundamental rule of differential calculus that prevents the incorrect assumption that the derivative of a product is simply the product of the derivatives. The product rule is used whenever functions are multiplied together and their rate of change is needed, for example in physics when computing power as the product of force and velocity.
Key Formula
d/dx[f(x) · g(x)] = f'(x) · g(x) + f(x) · g'(x)
LaTeX: \frac{d}{dx}[f(x) \cdot g(x)] = f'(x) \cdot g(x) + f(x) \cdot g'(x)
| Symbol | Meaning | Unit |
|---|---|---|
| f(x) | first differentiable function | dimensionless |
| g(x) | second differentiable function | dimensionless |
| f'(x) | derivative of the first function | dimensionless |
| g'(x) | derivative of the second function | dimensionless |
Worked Example
Problem
Differentiate h(x) = x³ · sin(x).
Solution
Step 1: Identify f(x) = x³ and g(x) = sin(x). Step 2: Find f'(x) = 3x². Step 3: Find g'(x) = cos(x). Step 4: Apply the product rule: h'(x) = f'(x)·g(x) + f(x)·g'(x). Step 5: h'(x) = 3x²·sin(x) + x³·cos(x).
Answer
h'(x) = 3x²sin(x) + x³cos(x)
Product Rule Applied to Common Function Pairs
| f(x) | g(x) | f'(x) | g'(x) | Derivative of f·g |
|---|---|---|---|---|
| x² | eˣ | 2x | eˣ | 2xeˣ + x²eˣ = eˣ(2x + x²) |
| x | ln x | 1 | 1/x | ln x + x · (1/x) = ln x + 1 |
| sin x | cos x | cos x | −sin x | cos²x − sin²x = cos(2x) |
| x³ | e^(2x) | 3x² | 2e^(2x) | 3x²e^(2x) + 2x³e^(2x) |
Interactive Tools
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Related Terms
Quotient Rule
The quotient rule provides a formula for finding the derivative of a function expressed as the ratio of two differentiable functions, stating that the derivative equals the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the square of the denominator. It is a direct consequence of the product rule applied to f(x) · [g(x)]⁻¹. The quotient rule is widely used in applied mathematics, physics, and engineering when dealing with rates expressed as fractions.
Chain Rule
The chain rule is a differentiation rule used to compute the derivative of a composite function, stating that the derivative of f(g(x)) equals the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. It is one of the most widely applied rules in calculus, essential whenever a function is "nested" inside another. The chain rule is critical in physics for related rates problems, in machine learning for backpropagation, and in multivariable calculus for total derivatives.
Differentiation
Differentiation is the process of computing the derivative of a function, yielding a new function that expresses the rate of change of the original at every point in its domain. It involves applying systematic rules — such as the power rule, product rule, chain rule, and quotient rule — to transform a given function into its derivative. Differentiation is used extensively in physics for velocity and acceleration, in economics for marginal analysis, and in engineering for optimisation and control systems.
The term "product rule" describes the rule for differentiating products of functions. It was discovered independently by Isaac Newton and Gottfried Leibniz in the 1670s as part of the foundational development of differential calculus.