Implicit or inverse differentiation

5. (a) Differentiate $$\frac { \cos 2 x } { \sqrt { x } }$$ with respect to \(x\).
(b) Show that \(\frac { \mathrm { d } } { \mathrm { d } x } \left( \sec ^ { 2 } 3 x \right)\) can be written in the form $$\mu \left( \tan 3 x + \tan ^ { 3 } 3 x \right)$$ where \(\mu\) is a constant.
(c) Given \(x = 2 \sin \left( \frac { y } { 3 } \right)\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\), simplifying your answer.

1. (i) Differentiate \(x ^ { 3 } \ln x\) with respect to \(x\).
   (ii) Given that

$$x = \frac { y + 1 } { 3 - 2 y }$$ find and simplify an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(y\).

9

1. Given that \(x = \frac { \sin y } { \cos y }\), use the quotient rule to show that $$\frac { \mathrm { d } x } { \mathrm {~d} y } = \sec ^ { 2 } y$$ (3 marks)
2. Given that \(\tan y = x - 1\), use a trigonometrical identity to show that $$\sec ^ { 2 } y = x ^ { 2 } - 2 x + 2$$
3. Show that, if \(y = \tan ^ { - 1 } ( x - 1 )\), then $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { x ^ { 2 } - 2 x + 2 }$$ (l mark)
4. A curve has equation \(y = \tan ^ { - 1 } ( x - 1 ) - \ln x\).
   1. Find the value of the \(x\)-coordinate of each of the stationary points of the curve.
   2. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
   3. Hence show that the curve has a minimum point which lies on the \(x\)-axis.

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

1. \(\sqrt { 1 - \cos x }\)
2. \(x ^ { 3 } \ln x\) (b) Given that $$x = \frac { y + 1 } { 3 - 2 y } ,$$ find and simplify an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(y\).

10 A curve has equation \(x = ( y + 5 ) \ln ( 2 y - 7 )\).

1. Find \(\frac { \mathrm { d } x } { \mathrm {~d} y }\) in terms of y .
2. Find the gradient of the curve where it crosses the y -axis.