For which values of n does the rule work at this level?

All rational n: positive, negative and fractional.

How do you differentiate \frac{x^2+4x+4}{\sqrt{x}}?

Divide each term by x^{\frac12} first, then differentiate term by term.

How do you find the gradient at x=a?

Substitute x=a into \frac{dy}{dx}.

Common error when differentiating x^3?

Writing x^2 without multiplying by 3.

Differentiating powers of xAQA A-Level Maths: Flashcards

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Question

State the rule for differentiating $x^n$.

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State the rule for differentiating $x^n$ .: $\frac{\mathrm{d}}{\mathrm{d}x}(x^n)=nx^{n-1}$
For which values of $n$ does the rule work at this level?: All rational $n$ : positive, negative and fractional.
Derivative of a constant?: $0$
Differentiate $kx^n$ .: $knx^{n-1}$
Differentiate $x$ .: $1$
Write $\sqrt{x}$ and $\frac{1}{\sqrt{x}}$ as powers.: $x^{\frac12}$ and $x^{-\frac12}$
Write $\frac{1}{x^3}$ as a power.: $x^{-3}$
Differentiate $5\sqrt{x}$ .: $\frac52x^{-\frac12}$
Differentiate $-12x^{-3}$ .: $36x^{-4}$
How do you differentiate $\frac{x^2+4x+4}{\sqrt{x}}$ ?: Divide each term by $x^{\frac12}$ first, then differentiate term by term.
How do you find the gradient at $x=a$ ?: Substitute $x=a$ into $\frac{\mathrm{d}y}{\mathrm{d}x}$ .
Equation of the tangent at $(a,y_1)$ ?: $y-y_1=\mathrm{f}'(a)(x-a)$
Common error when differentiating $x^3$ ?: Writing $x^2$ without multiplying by 3.

Exam questions on Differentiating powers of x

A curve has equation $y=4x^3-5x^2+7x-9$ .
Find the values of $x$ at which the gradient of the curve is 7.2 marks
A curve has equation $y=5\sqrt{x}-\frac{12}{x^3}$ , where $x>0$ .
Find the gradient of the curve at the point where $x=4$ .2 marks
The function $\mathrm{f}$ is defined by $\mathrm{f}(x)=\frac{(x+2)^2}{\sqrt{x}}$ , where $x>0$ .
Show that $\mathrm{f}(x)=x^{\frac32}+4x^{\frac12}+4x^{-\frac12}$ .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).