Scottish Higher Maths5 min read

Chain rule

Recognise composite functions and differentiate them using the chain rule, including coefficients, nonlinear inside functions, and negative or fractional powers.

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Composite functions

Some functions are built from one function applied to the output of another. In $y=(2x+5)^4$ , the expression $2x+5$ is worked out first, and the result is then raised to the power 4. A function built this way — a function of a function — is a composite function.

The chain rule

Worked example

Differentiate a bracket raised to a power

Differentiate $y=(2x+5)^4$ with respect to $x$ .

Step 1
Differentiate the outside function
Bring down the power and reduce it by 1, keeping the bracket unchanged: $4(2x+5)^3$ .
Step 2
Multiply by the derivative of the inside function
The derivative of $2x+5$ is $2$ , so multiply: $4(2x+5)^3\times2$ .

Final answer

$\frac{dy}{dx}=8(2x+5)^3$

The outside power is differentiated first, and the constant factor 2 comes from differentiating the inside function $2x+5$ .

Coefficients and nonlinear inside functions

An outer constant coefficient stays in place and is multiplied in at the end. The inside function does not have to be linear — it can be a quadratic or another expression in $x$ , and its own derivative is found in the usual way before multiplying.

Worked example

Coefficient with a nonlinear inside function

Differentiate $y=3(x^2+2)^4$ with respect to $x$ .

Step 1
Differentiate the outside function
Keep the coefficient 3, bring down the power 4 and reduce it by 1: $3\times4(x^2+2)^3$ .
Step 2
Multiply by the derivative of the inside function
The derivative of $x^2+2$ is $2x$ , so multiply: $12(x^2+2)^3\times2x$ .
Step 3
Combine the constants
$12\times2=24$ , giving $24x(x^2+2)^3$ .

Final answer

$\frac{dy}{dx}=24x(x^2+2)^3$

The coefficient 3 is carried through the whole calculation, and the inside function's own derivative ( $2x$ ) is what gets multiplied in — not just $x$ .

Dropping the outer coefficient 3, or multiplying by $x$ instead of the correct derivative $2x$.

Negative and fractional powers

The chain rule works the same way for negative and fractional outside powers. A square root can be rewritten as a power of $\frac12$ before differentiating, and a reciprocal can be rewritten as a negative power — after that, the same two steps apply: differentiate the outside power, then multiply by the derivative of the inside function.

Worked example

Differentiate a negative power

Differentiate $y=(3x-1)^{-2}$ with respect to $x$ .

Step 1
Differentiate the outside function
Bring down the power $-2$ and reduce it by 1, to $-3$ : $-2(3x-1)^{-3}$ .
Step 2
Multiply by the derivative of the inside function
The derivative of $3x-1$ is $3$ , so multiply: $-2(3x-1)^{-3}\times3$ .

Final answer

$\frac{dy}{dx}=-6(3x-1)^{-3}=-\frac{6}{(3x-1)^3}$

Reducing $-2$ by 1 gives $-3$ , not $-1$ — the power always reduces by exactly 1, whatever sign it starts with.

Reducing the power to $-1$ instead of $-3$.

Worked example

Differentiate a square root

Differentiate $y=\sqrt{4x+1}$ with respect to $x$ .

Step 1
Rewrite as a fractional power
$y=(4x+1)^{1/2}$ .
Step 2
Differentiate the outside function
Bring down the power $\frac12$ and reduce it by 1, to $-\frac12$ : $\frac12(4x+1)^{-1/2}$ .
Step 3
Multiply by the derivative of the inside function
The derivative of $4x+1$ is $4$ , so multiply: $\frac12(4x+1)^{-1/2}\times4$ .

Final answer

$\frac{dy}{dx}=2(4x+1)^{-1/2}=\frac{2}{\sqrt{4x+1}}$

Rewriting the square root as a power of $\frac12$ first turns this into an ordinary chain rule question.

Self-checkQuick self-check

Differentiate $y=2(3x^2+1)^3$ with respect to $x$ .

Answer

$\frac{dy}{dx}=36x(3x^2+1)^2$

Keep the coefficient 2, bring down the power 3 and reduce it to 2, then multiply by the derivative of $3x^2+1$ , which is $6x$ : $2\times3\times6=36$ .

Composite functions

The chain rule

Differentiate a bracket raised to a power

Differentiate the outside function

Multiply by the derivative of the inside function

Coefficients and nonlinear inside functions

Coefficient with a nonlinear inside function

Differentiate the outside function

Multiply by the derivative of the inside function

Combine the constants

Negative and fractional powers

Differentiate a negative power

Differentiate the outside function

Multiply by the derivative of the inside function

Differentiate a square root

Rewrite as a fractional power

Differentiate the outside function

Multiply by the derivative of the inside function