MathematicsCalculusA-Level
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Product Rule Calculator

Differentiating the product of two functions.

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Resultant Gradient

Formula first

Overview

The Product Rule is a fundamental differentiation formula used to find the derivative of a function that is the product of two or more differentiable functions. It establishes that the derivative of a product is not simply the product of the individual derivatives, but a specific combination of original functions and their respective rates of change.

Symbols

Variables

\frac{dy}{dx} = Resultant Gradient, u = Function u, \frac{dv}{dx} = Derivative v', v = Function v, \frac{du}{dx} = Derivative u'

Resultant Gradient
Function u
Derivative v'
Function v
Derivative u'

Apply it well

When To Use

When to use: Apply this rule when you encounter a function composed of two sub-functions multiplied together, such as algebraic, trigonometric, or exponential products. It is required when both factors in the product are non-constant functions of the same independent variable.

Why it matters: This rule is essential for calculating rates of change in systems with interacting variables, such as calculating the power in an electrical circuit (voltage times current) or the growth of economic revenue (price times quantity). It serves as the basis for the integration by parts method in integral calculus.

Avoid these traps

Common Mistakes

  • Just multiplying derivatives (u'v').
  • Sign errors.

One free problem

Practice Problem

A function is defined as the product of two sub-functions u and v. If u = 5 and v = 10, with their respective derivatives being du = 2 and dv = 4, calculate the total derivative dy.

Function u5
Function v10
Derivative u'2
Derivative v'4

Solve for:

Hint: Substitute the values into the formula: dy = (u × dv) + (v × du).

The full worked solution stays in the interactive walkthrough.

References

Sources

  1. Calculus by James Stewart
  2. Wikipedia: Product rule
  3. Stewart, James. Calculus: Early Transcendentals. 8th ed., Cengage Learning, 2016.
  4. Calculus: Early Transcendentals, 8th Edition by James Stewart
  5. Thomas' Calculus, 14th Edition by George B. Thomas Jr., Maurice D. Weir, Joel Hass
  6. Product rule (Wikipedia article title)
  7. Edexcel A-Level Mathematics — Pure (Differentiation)