Differentiation of reciprocal functions

8 \includegraphics[max width=\textwidth, alt={}, center]{de8af872-9f77-4787-8e66-ed199405ca25-3_581_650_1272_744} The diagram shows the curve $$y = \tan x \cos 2 x , \text { for } 0 \leqslant x < \frac { 1 } { 2 } \pi$$ and its maximum point \(M\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 4 \cos ^ { 2 } x - \sec ^ { 2 } x - 2\).
2. Hence find the \(x\)-coordinate of \(M\), giving your answer correct to 2 decimal places.

8 \includegraphics[max width=\textwidth, alt={}, center]{22ba6cc7-7375-434e-9eaa-d536684dd727-3_581_650_1272_744} The diagram shows the curve $$y = \tan x \cos 2 x , \text { for } 0 \leqslant x < \frac { 1 } { 2 } \pi$$ and its maximum point \(M\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 4 \cos ^ { 2 } x - \sec ^ { 2 } x - 2\).
2. Hence find the \(x\)-coordinate of \(M\), giving your answer correct to 2 decimal places.

8 \includegraphics[max width=\textwidth, alt={}, center]{9d3af34a-670f-425e-8156-0ad4d08fbdc0-3_499_552_258_792} The diagram shows the curve \(y = \operatorname { cosec } x\) for \(0 < x < \pi\) and part of the curve \(y = \mathrm { e } ^ { - x }\). When \(x = a\), the tangents to the curves are parallel.

1. By differentiating \(\frac { 1 } { \sin x }\), show that if \(y = \operatorname { cosec } x\) then \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \operatorname { cosec } x \cot x\).
2. By equating the gradients of the curves at \(x = a\), show that $$a = \tan ^ { - 1 } \left( \frac { \mathrm { e } ^ { a } } { \sin a } \right)$$
3. Verify by calculation that \(a\) lies between 1 and 1.5.
4. Use an iterative formula based on the equation in part (ii) to determine \(a\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.

8

1. By differentiating \(\frac { 1 } { \cos \theta }\), show that if \(y = \sec \theta\) then \(\frac { \mathrm { d } y } { \mathrm {~d} \theta } = \tan \theta \sec \theta\).
2. Hence show that $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} \theta ^ { 2 } } = a \sec ^ { 3 } \theta + b \sec \theta$$ giving the values of \(a\) and \(b\).
3. Find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \left( 1 + \tan ^ { 2 } \theta - 3 \sec \theta \tan \theta \right) d \theta$$

7 Let \(\mathrm { f } ( x ) = \frac { \cos x } { 1 + \sin x }\).

1. Show that \(\mathrm { f } ^ { \prime } ( x ) < 0\) for all \(x\) in the interval \(- \frac { 1 } { 2 } \pi < x < \frac { 3 } { 2 } \pi\).
2. Find \(\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 2 } \pi } \mathrm { f } ( x ) \mathrm { d } x\). Give your answer in a simplified exact form.

3. Given that $$y = \arctan \left( \frac { \sin x } { \cos x - 1 } \right) \quad x \neq 2 n \pi , \quad n \in \mathbb { Z }$$ Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = k$$ where \(k\) is a constant to be found. \(\_\_\_\_\) "

5. (i) Use the derivatives of \(\sin x\) and \(\cos x\) to prove that $$\frac { \mathrm { d } } { \mathrm {~d} x } ( \tan x ) = \sec ^ { 2 } x$$ The tangent to the curve \(y = 2 x \tan x\) at the point where \(x = \frac { \pi } { 4 }\) meets the \(y\)-axis at the point \(P\).
(ii) Find the \(y\)-coordinate of \(P\) in the form \(k \pi ^ { 2 }\) where \(k\) is a rational constant.

3

1. Show that the derivative of \(\sec x\) can be written as \(\sec x \tan x\).
2. Find \(\int \frac { \tan x } { \sqrt { 1 + \cos 2 x } } \mathrm {~d} x\).

1. Given that

$$x = \sec ^ { 2 } y + \tan y ,$$ show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { \cos ^ { 2 } y } { 2 \tan y + 1 } .$$

8

1. Given that \(y = \sec x\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \sec x \tan x\).
2. In this question you must show detailed reasoning. Find the exact value of \(\int _ { \frac { 1 } { 12 } \pi } ^ { \frac { 1 } { 6 } \pi } ( \sec 2 x + \tan 2 x ) ^ { 2 } \mathrm {~d} x\).

Use the derivatives of \(\sin x\) and \(\cos x\) to prove that the derivative of \(\tan x\) is \(\sec^2 x\). [4]