Product Rule
When you need to differentiate a function that is the product of two simpler functions, you cannot simply multiply their individual derivatives together. For example, if f(x) = x² and g(x) = sin(x), the derivative of their product f(x)·g(x) is not f'(x)·g'(x) = 2x·cos(x). Instead, calculus provides a specific rule—called the Product Rule—that correctly handles the differentiation of products. This rule is one of the foundational tools of differential calculus and appears constantly in physics, engineering, economics, and beyond. The Product Rule states that if two functions u(x) and v(x) are both differentiable at a point x, then the derivative of their product is u'(x)·v(x) + u(x)·v'(x). In words: 'the derivative of the first times the second, plus the first times the derivative of the second.' This formula can be extended to products of three or more functions, and it is closely related to the Quotient Rule for dividing functions. Understanding why the Product Rule works—not just how to apply it—deepens your intuition for calculus. The rule arises naturally from the limit definition of the derivative when you consider how a small change in x affects a product of two quantities that are both changing simultaneously. Mastering the Product Rule unlocks the ability to differentiate a vast class of functions, including polynomials multiplied by trigonometric, exponential, or logarithmic functions.
- Start from the limit definition
- Add and subtract a clever term
- Divide by Δx and take the limit
- Identify the two functions clearly
- Simplify the result
Start from the limit definition
The derivative of the product h(x) = u(x)·v(x) is defined as: h'(x) = lim[Δx→0] [u(x+Δx)·v(x+Δx) − u(x)·v(x)] / Δx. The trick is to add and subtract u(x+Δx)·v(x) in the numerator, which does not change the value but lets us factor the expression into two manageable pieces.
Add and subtract a clever term
Rewrite the numerator: u(x+Δx)·v(x+Δx) − u(x)·v(x) = u(x+Δx)·v(x+Δx) − u(x+Δx)·v(x) + u(x+Δx)·v(x) − u(x)·v(x). This groups as u(x+Δx)·[v(x+Δx)−v(x)] + v(x)·[u(x+Δx)−u(x)].
Divide by Δx and take the limit
Dividing by Δx gives: u(x+Δx)·[v(x+Δx)−v(x)]/Δx + v(x)·[u(x+Δx)−u(x)]/Δx. As Δx→0, u(x+Δx)→u(x), [v(x+Δx)−v(x)]/Δx → v'(x), and [u(x+Δx)−u(x)]/Δx → u'(x). Therefore h'(x) = u(x)·v'(x) + v(x)·u'(x).
Identify the two functions clearly
Before applying the rule, always label which part of the expression is u(x) and which is v(x). A good strategy is to choose u(x) as the function that simplifies when differentiated (often a polynomial) and v(x) as the function that stays manageable. Then compute u'(x) and v'(x) separately before substituting into the formula.
Simplify the result
After substituting into u'v + uv', collect like terms and simplify algebraically. Many errors occur in this final step, so be systematic: expand each product fully, then combine terms. For three functions u·v·w, apply the rule twice or use the extended form: (uvw)' = u'vw + uv'w + uvw'.
The Key Equation
- u(x) — First function (dimensionless (context-dependent)): The first differentiable factor in the product.
- v(x) — Second function (dimensionless (context-dependent)): The second differentiable factor in the product.
- u'(x) — Derivative of first function (per unit of x): The rate of change of u with respect to x, found using standard differentiation rules.
- v'(x) — Derivative of second function (per unit of x): The rate of change of v with respect to x, found using standard differentiation rules.
Worked Example
Problem. Find the derivative of h(x) = x³·eˣ.
Answer. h'(x) = x²eˣ(x + 3)
Origins · 1 minGottfried Wilhelm Leibniz and how this was discovered
Leibniz independently developed calculus around the same time as Isaac Newton and is credited with inventing the notation we use today, including the d/dx operator and the integral sign ∫. He derived the Product Rule and stated it clearly in his 1684 paper 'Nova Methodus,' making him the first to publish a systematic account of differential calculus. His notation proved far more practical than Newton's 'fluxion' notation and is universally used in modern mathematics. Leibniz was also a diplomat, philosopher, and inventor who designed an early mechanical calculator called the 'Stepped Reckoner' that could multiply and divide—centuries before electronic computers existed.
Related Concepts
- Quotient Rule — The Quotient Rule for differentiating f(x)/g(x) is derived directly from the Product Rule by writing the quotient as f(x)·[g(x)]⁻¹ and applying the chain rule alongside the product rule.
- Chain Rule — The Chain Rule handles composite functions and is frequently used together with the Product Rule when differentiating expressions like x²·sin(x³), where one factor is itself a composite function.
- Limit Definition of the Derivative — The Product Rule is rigorously proven using the limit definition of the derivative, so understanding limits is essential to fully appreciating why the rule works.
- Power Rule — The Power Rule is the most commonly used differentiation rule alongside the Product Rule; in most worked examples of the Product Rule, at least one of the two functions is differentiated using the Power Rule.