MathematicsCalculusA-Level

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Product Rule

Differentiating the product of two functions.

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\frac{d y}{d x} = u \frac{d v}{d x} + v \frac{d u}{d x}

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Core idea

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.

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.

Symbols

Variables

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

\frac{d y}{d x}

Resultant Gradient

V a r iab l e

u

Function u

V a r iab l e

\frac{d v}{d x}

Derivative v'

V a r iab l e

v

Function v

V a r iab l e

\frac{d u}{d x}

Derivative u'

V a r iab l e

Walkthrough

Derivation

Derivation of the Product Rule

The product rule differentiates the product of two functions u(x) and v(x). It is derived from first principles by adding and subtracting a convenient term.

u(x) and v(x) are differentiable.
The relevant limits exist.

Start with First Principles:

Apply the derivative definition to $f (x) = u (x) v (x)$ .

f^{'} (x) = h \to 0 lim \frac{u ( x + h ) v ( x + h ) - u ( x ) v ( x )}{h}

Add and Subtract u(x+h)v(x):

This changes the expression’s form without changing its value.

\frac{u ( x + h ) v ( x + h ) - u ( x + h ) v ( x ) + u ( x + h ) v ( x ) - u ( x ) v ( x )}{h}

Group and Factor:

Split into two difference quotients and factor out common terms.

h \to 0 lim [u (x + h) \frac{v ( x + h ) - v ( x )}{h} + v (x) \frac{u ( x + h ) - u ( x )}{h}]

Take the Limit:

As $h \to 0$ , $u (x + h) \to u (x)$ and the quotients become derivatives.

\frac{d}{d x} (uv) = u (x) v^{'} (x) + v (x) u^{'} (x)

Result

\frac{d}{d x} (uv) = u (x) v^{'} (x) + v (x) u^{'} (x)

Source: Edexcel A-Level Mathematics — Pure (Differentiation)

Free formulas

Rearrangements

Solve for $\frac{d y}{d x}$

Product Rule

\frac{d y}{d x} = u v^{'} + v u^{'}

This rearrangement demonstrates the common shorthand notation ( $u^{'}$ and $v^{'}$ ) for derivatives when applying the product rule.

Difficulty: 2/5

Solve for $u$

Make u the subject

u = \frac{\frac{d y}{d x} - v \frac{d u}{d x}}{\frac{d v}{d x}}

Isolate $u$ by subtracting the $v \frac{d u}{d x}$ term and dividing by $\frac{d v}{d x}$ .

Difficulty: 3/5

Solve for $v$

Make v the subject

v = \frac{\frac{d y}{d x} - u \frac{d v}{d x}}{\frac{d u}{d x}}

Isolate $v$ by subtracting the $u \frac{d v}{d x}$ term and dividing by $\frac{d u}{d x}$ .

Difficulty: 3/5

Solve for $\frac{d u}{d x}$

Make du/dx the subject

\frac{d u}{d x} = \frac{\frac{d y}{d x} - u \frac{d v}{d x}}{v}

Isolate $\frac{d u}{d x}$ by subtracting the $u \frac{d v}{d x}$ term and dividing by $v$ .

Difficulty: 2/5

Solve for $\frac{d v}{d x}$

Make dv/dx the subject

\frac{d v}{d x} = \frac{\frac{d y}{d x} - v \frac{d u}{d x}}{u}

Isolate $\frac{d v}{d x}$ by subtracting the $v \frac{d u}{d x}$ term and dividing by $u$ .

Difficulty: 2/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Visual intuition

Graph

Graph unavailable for this formula.

The product rule describes the rate of change of a product of two functions, meaning the graph typically represents the transformation of a composite product curve (f(x)g(x)) and its derivative (f'(x)g(x) + f(x)g'(x)). The curve shape varies depending on the component functions, but it generally depicts how the tangent slope of the product function is composed of the combined rates of change of its individual factors. This relationship visualizes how two independent variables interact to dictate the instantaneous growth or decay of their product.

Graph type: polynomial

Why it behaves this way

Intuition

Imagine a rectangle whose side lengths are functions of an independent variable; the rate of change of its area is the sum of the rate at which its width changes (scaled by its current height)

dy/dx

The instantaneous rate of change of the product of two functions, u and v, with respect to the independent variable x.

How quickly the overall quantity represented by the product u*v is increasing or decreasing as x changes.

The value of the first function at a specific point x.

The current 'size' or 'contribution' of the first factor to the product.

The value of the second function at a specific point x.

The current 'size' or 'contribution' of the second factor to the product.

du/dx

The instantaneous rate of change of the first function u with respect to x.

How quickly the first factor u is changing as x changes, independent of v.

dv/dx

The instantaneous rate of change of the second function v with respect to x.

How quickly the second factor v is changing as x changes, independent of u.

Signs and relationships

+: The total rate of change of the product is the sum of two distinct contributions: the rate of change of v scaled by u, and the rate of change of u scaled by v.

Free study cues

Insight

Canonical usage

The Product Rule ensures dimensional consistency when differentiating a function that is the product of two other functions, where the units of the derivative are the units of the product of the functions divided by the

Common confusion

A common mistake is assuming that the derivative of a product is the product of the derivatives, i.e., d(uv)/dx = (du/dx)(dv/dx). This is incorrect both mathematically and dimensionally, as the units of (du/dx)(dv/dx)

Unit systems

$x$ Any unit appropriate for the independent variable (e.g., s, m, dimensionless) · The independent variable.

$y$ Any unit appropriate for the dependent variable (e.g., m, J, dimensionless) · The dependent variable, where y = u*v. Its dimension is the product of the dimensions of u and v, i.e., [U][V].

$u$ Any unit appropriate for the first function (e.g., m, V, dimensionless) · The first differentiable function of x.

$v$ Any unit appropriate for the second function (e.g., s, A, dimensionless) · The second differentiable function of x.

$d y / d x$ Units of y / Units of x (e.g., m/s, J/m) · The derivative of y with respect to x. Dimensionally, this is equivalent to [U][V]/[X].

$d u / d x$ Units of u / Units of x (e.g., m/s, V/m) · The derivative of u with respect to x.

$d v / d x$ Units of v / Units of x (e.g., s/s, A/m) · The derivative of v with respect to x.

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: $d y$

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

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

Dampened harmonic motion (e^-x * sinx).

Study smarter

Tips

Explicitly label u and v before differentiating.
Calculate du and dv separately to avoid algebraic errors.
Remember that the order of the two added terms does not matter.
Use parentheses when substituting expressions to keep signs correct.

Avoid these traps

Common Mistakes

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

Common questions

Frequently Asked Questions

The product rule differentiates the product of two functions u(x) and v(x). It is derived from first principles by adding and subtracting a convenient term.

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.

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.

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

Dampened harmonic motion (e^-x * sinx).

Explicitly label u and v before differentiating. Calculate du and dv separately to avoid algebraic errors. Remember that the order of the two added terms does not matter. Use parentheses when substituting expressions to keep signs correct.

References

Sources

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

Product Rule

Overview

Variables

Derivation

Start with First Principles:

Add and Subtract u(x+h)v(x):

Group and Factor:

Take the Limit:

Rearrangements

Graph

Intuition

Insight

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Quotient Rule

Frequently Asked Questions

Sources