Show derivative equals expression - algebraic/trigonometric identity proof

Show that dy/dx equals a specific simplified expression, requiring manipulation through product/quotient/chain rule and then algebraic or trigonometric identity simplification to reach the target form.

6 questions · Standard +0.3

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CAIE P2 2022 March Q5

8 marks Standard +0.3

Given that $y = \tan ^ { 2 } x$, show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 \tan x + 2 \tan ^ { 3 } x$.
Find the exact value of $\int _ { \frac { 1 } { 4 } \pi } ^ { \frac { 1 } { 3 } \pi } \left( \tan x + \tan ^ { 2 } x + \tan ^ { 3 } x \right) \mathrm { d } x$.

OCR MEI C3 2013 June Q5

4 marks Moderate -0.3

5 Given that $y = \ln \left( \sqrt { \frac { 2 x - 1 } { 2 x + 1 } } \right)$, show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 2 x - 1 } - \frac { 1 } { 2 x + 1 }$.

Edexcel PMT Mocks Q15

6 marks Standard +0.3

The curve $C$ has equation

$$y = \frac { 1 } { 2 } x - \frac { 1 } { 4 } \sin 2 x \quad 0 < x < \pi$$ a. Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \sin ^ { 2 } x$ b. Find the coordinates of the points of inflection of the curve.

AQA C3 2013 January Q3

7 marks Moderate -0.3

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ when $$y = \mathrm { e } ^ { 3 x } + \ln x$$
1. Given that $u = \frac { \sin x } { 1 + \cos x }$, show that $\frac { \mathrm { d } u } { \mathrm {~d} x } = \frac { 1 } { 1 + \cos x }$.
2. Hence show that if $y = \ln \left( \frac { \sin x } { 1 + \cos x } \right)$, then $\frac { \mathrm { d } y } { \mathrm {~d} x } = \operatorname { cosec } x$.

AQA Paper 2 Specimen Q8

8 marks Standard +0.8

A curve has equation $y = 2x \cos 3x + (3x^2 - 4) \sin 3x$

Find $\frac{dy}{dx}$, giving your answer in the form $(mx^2 + n) \cos 3x$, where $m$ and $n$ are integers. [4 marks]
Show that the $x$-coordinates of the points of inflection of the curve satisfy the equation $$\cot 3x = \frac{9x^2 - 10}{6x}$$ [4 marks]

SPS SPS FM Pure 2025 June Q4

5 marks Standard +0.8

Given that $$y = \frac{3\sin \theta}{2\sin \theta + 2\cos \theta} \quad -\frac{\pi}{4} < \theta < \frac{3\pi}{4}$$ show that $$\frac{dy}{d\theta} = \frac{A}{1 + \sin 2\theta} \quad -\frac{\pi}{4} < \theta < \frac{3\pi}{4}$$ where $A$ is a rational constant to be found. [5]