MathematicsCalculusA-Level

IEBEdexcelAQACCEAOCRWJECAPIB

Chain Rule

Differentiating composite functions.

Understand the formulaSee the free derivationOpen the full walkthrough

\frac{d y}{d x} = \frac{d y}{d u} \times \frac{d u}{d x}

Open Full Walkthrough Try Calculator

This public page keeps the free explanation visible and leaves premium worked solving, advanced walkthroughs, and saved study tools inside the app.

Core idea

Overview

The Chain Rule is a fundamental formula in calculus used to find the derivative of a composite function. It establishes that the derivative of a nested function is the product of the derivative of the outer function and the derivative of the inner function.

When to use: Apply this rule when you need to differentiate a function that is composed of other functions, often described as a function within a function. It is necessary for expressions involving powers of polynomials, trigonometric functions with complex arguments, or exponential functions where the exponent is another function.

Why it matters: This rule is the foundation for many advanced mathematical concepts, including the backpropagation algorithm used to train neural networks in artificial intelligence. In physics and engineering, it allows for the analysis of related rates, such as how the volume of a sphere changes over time as its radius expands.

Symbols

Variables

\frac{dy}{dx} = Total Derivative, \frac{dy}{du} = Outer Derivative, \frac{du}{dx} = Inner Derivative

\frac{d y}{d x}

Total Derivative

V a r iab l e

\frac{d y}{d u}

Outer Derivative

V a r iab l e

\frac{d u}{d x}

Inner Derivative

V a r iab l e

Walkthrough

Derivation

Understanding the Chain Rule

The chain rule differentiates a composite function by multiplying the derivative of the outer function by the derivative of the inner function.

y depends on u and u depends on x.
Both functions are differentiable.

Introduce an Inner Variable:

Represent the inner function as u to separate the composite into two steps.

y = f (g (x)), let u = g (x) ⟹ y = f (u)

State the Chain Rule:

Differentiate outer with respect to u, then multiply by the derivative of u with respect to x.

\frac{d y}{d x} = \frac{d y}{d u} \cdot \frac{d u}{d x}

Note: A useful intuition is that changes in x affect u, which then affects y, so the overall rate multiplies.

Result

\frac{d y}{d x} = \frac{d y}{d u} \cdot \frac{d u}{d x}

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

Free formulas

Rearrangements

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

Make dydx the subject

\frac{d y}{d x} = \frac{d y}{d u} \times \frac{d u}{d x}

The dydx is already the subject of the formula.

Difficulty: 1/5

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

Make dydu the subject

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

Rearrange the chain rule formula to solve for dydu.

Difficulty: 2/5

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

Make dudx the subject

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

Rearrange the chain rule formula to solve for dudx.

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 graph of the Chain Rule is not a single static curve, but rather a representation of the relationship between composite functions where the slope of the composite function is the product of the slopes of the inner and outer functions. At any given point on the graph, the steepness of the composite curve is determined by multiplying the rate of change of the outer function by the derivative of the inner function. This visualizes how the composition of two functions scales the input variable's influence on the final output.

Graph type: polynomial

Why it behaves this way

Intuition

Imagine a chain of events where a small change in 'x' first causes a scaled change in 'u', and then that change in 'u' further causes another scaled change in 'y', with the total change in 'y' relative to 'x' being the

dy/dx

The instantaneous rate at which the output variable 'y' changes with respect to a change in the independent variable 'x'.

How sensitive 'y' is to changes in 'x' directly, representing the overall rate of change.

dy/du

The instantaneous rate at which the outer function's output 'y' changes with respect to a change in its immediate input 'u'.

How sensitive the 'outer layer' of the function is to changes in what it directly receives as input.

du/dx

The instantaneous rate at which the intermediate function's output 'u' changes with respect to a change in its input 'x'.

How sensitive the 'inner layer' of the function is to changes in its own input 'x'.

Free study cues

Insight

Canonical usage

Used to ensure dimensional consistency when calculating derivatives of composite functions, where the units of the derivative of the outer function multiplied by the units of the derivative of the inner function must

Common confusion

A common mistake is neglecting to consider the units of the intermediate variable 'u', which can lead to an incorrect understanding of the units for the final derivative.

Unit systems

$d y / d x$ [unit of y] / [unit of x] · The unit of a derivative dy/dx is the unit of the dependent variable y divided by the unit of the independent variable x. The Chain Rule requires that the units of the product (dy/du) * (du/dx)

$d y / d u$ [unit of y] / [unit of u] · Represents the rate of change of y with respect to u.

$d u / d x$ [unit of u] / [unit of x] · Represents the rate of change of u with respect to x.

One free problem

Practice Problem

In a calculus exercise involving a composite function, you determine that the derivative of the outer function with respect to its internal variable is 5, and the derivative of that internal variable with respect to x is 4. Calculate the total derivative dy/dx.

Outer Derivative5

Inner Derivative4

Solve for: $d y d x$

Hint: The Chain Rule states that the total derivative is the product of the outer and inner derivatives.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

Rate of reaction dependent on temperature dependent on time.

Study smarter

Tips

Identify the 'inner' function (u) and 'outer' function (y) clearly before starting.
Differentiate the outer layer while keeping the inner layer unchanged, then multiply by the inner derivative.
Work systematically from the outermost layer to the innermost layer in nested composites.

Avoid these traps

Common Mistakes

Forgetting the inner derivative.
Adding instead of multiplying.

Common questions

Frequently Asked Questions

The chain rule differentiates a composite function by multiplying the derivative of the outer function by the derivative of the inner function.

Apply this rule when you need to differentiate a function that is composed of other functions, often described as a function within a function. It is necessary for expressions involving powers of polynomials, trigonometric functions with complex arguments, or exponential functions where the exponent is another function.

This rule is the foundation for many advanced mathematical concepts, including the backpropagation algorithm used to train neural networks in artificial intelligence. In physics and engineering, it allows for the analysis of related rates, such as how the volume of a sphere changes over time as its radius expands.

Forgetting the inner derivative. Adding instead of multiplying.