Find derivative of quotient

4

1. Differentiate \(\frac { 2 x ^ { 3 } + 5 } { x }\) with respect to \(x\).
2. Find \(\int ( 3 x - 2 ) ^ { 5 } \mathrm {~d} x\) and hence find the value of \(\int _ { 0 } ^ { 1 } ( 3 x - 2 ) ^ { 5 } \mathrm {~d} x\).

1 Given that \(y = \frac { \ln x } { x ^ { 2 } }\), find the exact value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(x = \mathrm { e }\).

2 Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in each of the following cases:

1. \(y = \ln ( 1 + \sin 2 x )\),
2. \(y = \frac { \tan x } { x }\).

4. (a) Differentiate with respect to \(x\)

1. \(x ^ { 2 } \mathrm { e } ^ { 3 x + 2 }\),
2. \(\frac { \cos \left( 2 x ^ { 3 } \right) } { 3 x }\).
   (b) Given that \(x = 4 \sin ( 2 y + 6 )\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\).

4. (i) Given that \(y = \frac { \ln \left( x ^ { 2 } + 1 \right) } { x }\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
(ii) Given that \(x = \tan y\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 1 + x ^ { 2 } }\).

Differentiate with respect to \(x\), giving your answer in its simplest form,

1. \(x ^ { 2 } \ln ( 3 x )\)
2. \(\frac { \sin 4 x } { x ^ { 3 } }\)

5. (i) Differentiate with respect to \(x\)

1. \(y = x ^ { 3 } \ln 2 x\)
2. \(y = ( x + \sin 2 x ) ^ { 3 }\) Given that \(x = \cot y\),
   (ii) show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { - 1 } { 1 + x ^ { 2 } }\)

1. (i) Given that

$$x = \sec ^ { 2 } 2 y , \quad 0 < y < \frac { \pi } { 4 }$$ show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 4 x \sqrt { ( x - 1 ) } }$$ (ii) Given that $$y = \left( x ^ { 2 } + x ^ { 3 } \right) \ln 2 x$$ find the exact value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at \(x = \frac { \mathrm { e } } { 2 }\), giving your answer in its simplest form.
(iii) Given that $$f ( x ) = \frac { 3 \cos x } { ( x + 1 ) ^ { \frac { 1 } { 3 } } } , \quad x \neq - 1$$ show that $$\mathrm { f } ^ { \prime } ( x ) = \frac { \mathrm { g } ( x ) } { ( x + 1 ) ^ { \frac { 4 } { 3 } } } , \quad x \neq - 1$$ where \(\mathrm { g } ( x )\) is an expression to be found.

4. Differentiate with respect to \(x\)

1. \(x ^ { 3 } \mathrm { e } ^ { 3 x }\),
2. \(\frac { 2 x } { \cos x }\),
3. \(\tan ^ { 2 } x\). Given that \(x = \cos y ^ { 2 }\),
4. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(y\).

3. Differentiate each of the following with respect to \(x\) and simplify your answers.

1. \(\quad \ln ( 3 x - 2 )\)
2. \(\frac { 2 x + 1 } { 1 - x }\)
3. \(x ^ { \frac { 3 } { 2 } } \mathrm { e } ^ { 2 x }\)

2 Given that \(y = \frac { \cos x } { 1 - \sin x }\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), simplifying your answer.

6

1. Use the quotient rule to show that the derivative of \(\frac { \cos x } { \sin x }\) is \(\frac { - 1 } { \sin ^ { 2 } x }\).
2. Show that \(\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 4 } \pi } \frac { \sqrt { 1 + \cos 2 x } } { \sin x \sin 2 x } \mathrm {~d} x = \frac { 1 } { 2 } ( \sqrt { 6 } - \sqrt { 2 } )\).

1 Differentiate the following with respect to \(x\).

1. \(\mathrm { e } ^ { - 4 x }\)
2. \(\frac { x ^ { 2 } } { x + 1 }\)

4 Differentiate the following.

1. \(\sqrt { 1 - 3 x ^ { 2 } }\)
2. \(\frac { x ^ { 2 } } { 3 x + 2 }\)

2

1. Given that \(y = x ^ { 4 } \tan 2 x\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
   (3 marks)
2. Find the gradient of the curve with equation \(y = \frac { x ^ { 2 } } { x - 1 }\) at the point where \(x = 3\).
   (3 marks)

4. Differentiate each of the following with respect to \(x\) and simplify your answers.

1. \(\quad \ln ( 3 x - 2 )\)
2. \(\frac { 2 x + 1 } { 1 - x }\)
3. \(x ^ { \frac { 3 } { 2 } } \mathrm { e } ^ { 2 x }\)

1

1. Differentiate the following with respect to \(x\).
   1. \(\frac { 1 } { ( 3 x - 4 ) ^ { 2 } }\)
   2. \(\frac { \ln ( x + 2 ) } { x }\)
   3. Find \(\int \mathrm { e } ^ { ( 2 x + 3 ) } \mathrm { d } x\).

5

1. By writing \(\tan x\) as \(\frac { \sin x } { \cos x }\), use the quotient rule to show that \(\frac { \mathrm { d } } { \mathrm { d } x } ( \tan x ) = \sec ^ { 2 } x\).
   [0pt] [2 marks]
2. Use integration by parts to find \(\int x \sec ^ { 2 } x \mathrm {~d} x\).
   [0pt] [4 marks]
3. The region bounded by the curve \(y = ( 5 \sqrt { x } ) \sec x\), the \(x\)-axis from 0 to 1 and the line \(x = 1\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid. Find the value of the volume of the solid generated, giving your answer to two significant figures.
   [0pt] [3 marks]

10

1. Given that $$y = \tan x$$ use the quotient rule to show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \sec ^ { 2 } x$$ 10
2. The region enclosed by the curve \(y = \tan ^ { 2 } x\) and the horizontal line, which intersects the curve at \(x = - \frac { \pi } { 4 }\) and \(x = \frac { \pi } { 4 }\), is shaded in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{042e248a-9efa-4844-957d-f05715900ffc-17_1059_967_461_539} Show that the area of the shaded region is $$\pi - 2$$ Fully justify your answer.
   Do not write outside the box \includegraphics[max width=\textwidth, alt={}, center]{042e248a-9efa-4844-957d-f05715900ffc-19_2488_1716_219_153}

Differentiate with respect to \(x\) $$\frac{6x^2 - 1}{2\sqrt{x}}.$$ [5]

Given that $$y = \frac{x^3}{\sin x}$$ find \(\frac{dy}{dx}\) [3 marks]