Differentiate inverse trigonometric functions

1 Given that \(y = \sin ^ { - 1 } \left( x ^ { 2 } \right)\), find \(\frac { d y } { d x }\).

3

1. Given that \(x = \tan ( 3 y + 1 )\) :
   1. find \(\frac { \mathrm { d } x } { \mathrm {~d} y }\) in terms of \(y\);
   2. find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(y = - \frac { 1 } { 3 }\).
2. Sketch the graph of \(y = \tan ^ { - 1 } x\).

9 The diagram shows the curve with equation \(y = \sin ^ { - 1 } 2 x\), where \(- \frac { 1 } { 2 } \leqslant x \leqslant \frac { 1 } { 2 }\).
\includegraphics[max width=\textwidth, alt={}, center]{0ab0e757-270b-4c15-9202-9df2f02dddf3-4_790_752_906_644}

1. Find the \(y\)-coordinate of the point \(A\), where \(x = \frac { 1 } { 2 }\).
2. 1. Given that \(y = \sin ^ { - 1 } 2 x\), show that \(x = \frac { 1 } { 2 } \sin y\).
   2. Given that \(x = \frac { 1 } { 2 } \sin y\), find \(\frac { \mathrm { d } x } { \mathrm {~d} y }\) in terms of \(y\).
3. Using the answers to part (b) and a suitable trigonometrical identity, show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 } { \sqrt { 1 - 4 x ^ { 2 } } }$$

2 In this question you must show detailed reasoning.
Find the gradient of the curve \(y = 6 \arcsin ( 2 x )\) at the point with \(x\)-coordinate \(\frac { 1 } { 4 }\). Express the result in the form \(\mathrm { m } \sqrt { \mathrm { n } }\), where \(m\) and \(n\) are integers.

11. (a) Differentiate each of the following with respect to \(x\).

1. \(y = \mathrm { e } ^ { 3 x } \sin ^ { - 1 } x\)
2. \(y = \ln \left( \cosh ^ { 2 } \left( 2 x ^ { 2 } + 7 x \right) \right)\)
   (b) Find the equations of the tangents to the curve \(x = \sinh ^ { - 1 } \left( y ^ { 2 } \right)\) at the points where \(x = 1\).

10. (a) By writing \(y = \sin ^ { - 1 } ( 2 x + 5 )\) as \(\sin y = 2 x + 5\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 } { \sqrt { 1 - ( 2 x + 5 ) ^ { 2 } } }\).
(b) Deduce the range of values of \(x\) for which \(\frac { \mathrm { d } } { \mathrm { d } x } \left( \sin ^ { - 1 } ( 2 x + 5 ) \right)\) is valid.

1. (a) Given that

$$y = \arcsin x \quad - 1 \leqslant x \leqslant 1$$ show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \sqrt { 1 - x ^ { 2 } } }$$ (b) $$\mathrm { f } ( x ) = \arcsin \left( \mathrm { e } ^ { x } \right) \quad x \leqslant 0$$ Prove that \(\mathrm { f } ( x )\) has no stationary points.