Differentiate \frac{1}{x^{2}}.

Write as x^{-2}, giving -2x^{-3}.

How do you differentiate (2x+5)(x-1)?

Expand to 2x^{2}+3x-5 first, then differentiate: 4x+3.

What if f''(x)=0 at a stationary point?

The test is inconclusive; check the sign of f'(x) either side.

Steps in an optimisation problem?

Express in one variable, differentiate, set to 0, solve, justify max/min, answer.

Differentiate \frac{x^{2}+3x-5}{4x^{1/2}}.

Split to \frac14x^{3/2}+\frac34x^{1/2}-\frac54x^{-1/2}, giving \frac38x^{1/2}+\frac38x^{-1/2}+\frac58x^{-3/2}.

Differentiating powers of x and stationary pointsEdexcel A-Level Maths: Flashcards

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Differentiate $kx^{n}$.

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Differentiate $kx^{n}$ .: $knx^{n-1}$
Differentiate $\sqrt{x}$ .: $\frac12x^{-1/2}$
Differentiate $\frac{1}{x^{2}}$ .: Write as $x^{-2}$ , giving $-2x^{-3}$ .
How do you differentiate $(2x+5)(x-1)$ ?: Expand to $2x^{2}+3x-5$ first, then differentiate: $4x+3$ .
Equation of the tangent at $x=a$ ?: $y-f(a)=f'(a)(x-a)$
Gradient of the normal if the tangent has gradient $m$ ?: $-\frac1m$
What is a stationary point?: A point where $f'(x)=0$ .
What does $f''(x)>0$ at a stationary point show?: A minimum.
What does $f''(x)<0$ at a stationary point show?: A maximum.
When is a function increasing?: When $f'(x)>0$ .
What if $f''(x)=0$ at a stationary point?: The test is inconclusive; check the sign of $f'(x)$ either side.
Steps in an optimisation problem?: Express in one variable, differentiate, set to $0$ , solve, justify max/min, answer.
Differentiate $\frac{x^{2}+3x-5}{4x^{1/2}}$ .: Split to $\frac14x^{3/2}+\frac34x^{1/2}-\frac54x^{-1/2}$ , giving $\frac38x^{1/2}+\frac38x^{-1/2}+\frac58x^{-3/2}$ .

Exam questions on Differentiating powers of x and stationary points

A curve has equation $y=(2x+5)(x-1)$ .
Find the $x$ -coordinate of the stationary point of the curve.2 marks
The function $f$ is defined by $f(x)=\frac{x^{2}+3x-5}{4x^{1/2}}$ for $x>0$ .
Find the exact value of $f'(4)$ .2 marks
A curve has equation $y=x^{3}-6x^{2}+9x+2$ .
Find the coordinates of the stationary points of the curve.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).