A-level · Differentiation

Differentiation from first principlesExam-style questions

The definition every differentiation rule is built on. What "the gradient at a point" means, as the limit of chords; the definition f'(x) as a limit; writing out f(x + h) correctly; cancelling h before letting it shrink; setting out the proof the way the marks are given; and finding the gradient at one point from first principles.

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8 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 2 Level 3 Level 4 Level 5

1
Given that

$y = 5x^2 - 3x$

use differentiation from first principles to show that

$\frac{dy}{dx} = 10x - 3$

(3 marks)
Level 2 · 3 marksTry it
2
A curve has equation $y = \mathrm{f}(x)$ , where $\mathrm{f}(x) = 2\sqrt{x} + \dfrac{16}{x}$ for $x > 0$ . The points $P(4,\ 8)$ , $Q(3.999,\ k)$ , $R(3.99,\ 8.00502)$ , $S(3.9,\ 8.05225)$ and $T(3,\ 8.79743)$ lie on the curve. Values that are not exact are given correct to 5 decimal places.
(a) Find the value of $k$ . Give your answer correct to 5 decimal places.

(1 mark)
(b) The table shows the gradients of the chords $PT$ , $PS$ , $PR$ and $PQ$ , correct to 4 decimal places.
Chord $PT$ $PS$ $PR$ $PQ$
Gradient of chord $-0.7974$ $-0.5022$ $-0.5002$
Find the gradient of the chord $PS$ . Give your answer correct to 4 decimal places.

(1 mark)
(c) State what the values in the table indicate about the value of $\mathrm{f}'(4)$ .

(1 mark)
Level 2 · 3 marksTry it
3
Given that

$f(x) = 2x^3 - 5x$ ,

use differentiation from first principles to show that $f'(x) = 6x^2 - 5$ . (4 marks)
Level 3 · 4 marksTry it
4
Differentiate $\sin x$ with respect to $x$ from first principles.

(4 marks)
Level 3 · 4 marksTry it
5
Prove, from first principles, that the derivative of $\sin x$ is $\cos x$ .

You may assume the addition formula for $\sin(A + B)$ and the small-angle limits $\dfrac{\sin h}{h} \to 1$ and $\dfrac{\cos h - 1}{h} \to 0$ as $h \to 0$ . (5 marks)
Level 3 · 5 marksTry it

Chord	$PT$	$PS$	$PR$	$PQ$
Gradient of chord	$-0.7974$		$-0.5022$	$-0.5002$

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