A-level · Differentiation

Differentiating trigonometric functionsExam-style questions

How fast a sine or cosine is changing. Why the gradient of sin x traces out cos x; proving it from first principles with the small-angle results; why x must be in radians; the other four derivatives from the quotient rule, and the minus every co-function carries; the chain rule on sin kx and sin²x; and using trig derivatives for rates, tangents and stationary points.

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19 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
Find $\dfrac{dy}{dx}$ for
(a) $y = \sec x$ and (b) $y = \cot x$ , deriving each from the quotient rule.
Level 3 · 4 marksTry it
2
A curve has equation $y = x\sin x$ . Show that $\dfrac{dy}{dx} = \sin x + x\cos x$ , and find the gradient at the point where $x = \dfrac{\pi}{2}$ .
Level 3 · 3 marksTry it
3
Differentiate $y = \cos^{3} x$ . (3 marks)
Level 3 · 3 marksTry it
4
Differentiate $y = x^2\cos x$ .
Level 3 · 2 marksTry it
5
The curve $C$ has equation $y = 3\sin 2x + \cos 2x$ .
(a) Find $\dfrac{dy}{dx}$ . (2 marks)
(b) Find an equation of the tangent to $C$ at the point where $x = \dfrac{\pi}{4}$ . (3 marks)
(c) This tangent crosses the $x$ -axis at the point $P$ . Find the exact $x$ -coordinate of $P$ . (2 marks)
Level 3 · 7 marksTry it

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