Differentiating powers of xEdexcel International A Level Maths: Revision notes

Section 1

The rule for differentiating powers of x

If $y=x^n$ , then $\frac{\mathrm{d}y}{\mathrm{d}x}=nx^{n-1},$ for any rational $n$ (positive, negative or fractional). Multiply by the power, then reduce the power by one. Special cases: $x$ differentiates to $1$ , and any constant differentiates to $0$ . Examples: $x^5\to5x^4$ ; $x^{-2}\to-2x^{-3}$ ; $x^{\frac12}\to\frac12x^{-\frac12}$ ; $x^{\frac32}\to\frac32x^{\frac12}$ .

Key termspower rule

Common mistake

Multiplying by the power but forgetting to subtract one from it, for example $x^4\to4x^4$ .

Section 2

Sums, differences and constant multiples

Differentiate term by term. A constant multiple stays in place: $\frac{\mathrm{d}}{\mathrm{d}x}\left(ax^n\right)=anx^{n-1}.$ Sums and differences are differentiated separately. For $y=3x^4-2x^2+7x-5$ : $\frac{\mathrm{d}y}{\mathrm{d}x}=12x^3-4x+7.$ The gradient at $x=1$ is $12-4+7=15$ , and the gradient where the curve meets the $y$ -axis ( $x=0$ ) is $7$ .

Key termsterm by term

Common mistake

Differentiating $7x$ as $7x$ . It becomes $7$ , because $x$ differentiates to $1$ .

Section 3

Rewriting before differentiating

The power rule needs a single power of $x$ . Rewrite roots and reciprocals first: $\sqrt{x}=x^{\frac12},\quad\frac{1}{x}=x^{-1},\quad\frac{1}{x^n}=x^{-n},\quad\frac{1}{\sqrt{x}}=x^{-\frac12},\quad\sqrt[3]{x}=x^{\frac13}.$ Example: $y=4\sqrt{x}+\frac{6}{x^2}=4x^{\frac12}+6x^{-2}$ , so $\frac{\mathrm{d}y}{\mathrm{d}x}=2x^{-\frac12}-12x^{-3}$ . You may give the answer in either index form or as a fraction, for example $2x^{-\frac12}-\frac{12}{x^3}$ , unless the question asks otherwise.

Key termsindex form

Common mistake

Writing $\frac{6}{x^2}$ as $(6x)^{-2}$ . Only $x$ is raised to the power: $6x^{-2}$ .

Exam tip

Differentiate negative and fractional powers in the same way as positive ones; the sign of $n$ is kept in $n x^{n-1}$ .

Section 4

Expanding brackets and dividing first

If the function is a product of brackets, expand it before differentiating. For $f(x)=(2x+5)(x-1)=2x^2+3x-5$ , $f'(x)=4x+3$ . If the function is a fraction with a single term in the denominator, divide each term by it. For $y=\frac{x^2+5x-3}{3x}$ : $y=\frac{x}{3}+\frac53-x^{-1}\quad\Rightarrow\quad\frac{\mathrm{d}y}{\mathrm{d}x}=\frac13+x^{-2}.$ For $f(x)=\frac{(x+2)^2}{x}=x+4+4x^{-1}$ , $f'(x)=1-4x^{-2}$ . Differentiating each bracket and multiplying the results is not a valid method.

Key termsexpand

Common mistake

Treating $\frac{x^2+5x-3}{3x}$ as $\frac{x^2}{3x}+5x-3$ : the denominator divides every term.

Exam tip

Once simplified, use the derivative to find gradients or to solve $\frac{\mathrm{d}y}{\mathrm{d}x}=k$ , for example finding where the gradient is $\frac43$ .

That's the notes covered.

Carry on to the next subtopic.

Exam questions on Differentiating powers of x

A curve has equation $y=3x^4-2x^2+7x-5$ .
Find the gradient of the curve at the point where it crosses the $y$ -axis.2 marks
A curve has equation $y=f(x)$ , where $f(x)=(2x+5)(x-1)$ .
The curve crosses the positive $x$ -axis at the point $A$ . Find the gradient of the curve at $A$ .2 marks
A curve has equation $y=\dfrac{x^2+5x-3}{3x}$ for $x\neq0$ .
Find $\frac{\mathrm{d}y}{\mathrm{d}x}$ .3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).