Differentiate inverse trigonometric functions

Find the derivative of functions involving arcsin, arccos, arctan, or their compositions with other functions.

10 questions · Standard +0.4

Sort by: Default | Easiest first | Hardest first
OCR Further Pure Core 1 2024 June Q1
3 marks Moderate -0.5
1 Given that \(y = \sin ^ { - 1 } \left( x ^ { 2 } \right)\), find \(\frac { d y } { d x }\).
AQA C3 2011 January Q3
6 marks Moderate -0.3
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 \(y = \tan ^ { - 1 } x\).
AQA C3 2006 June Q9
7 marks Standard +0.3
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 \(x = \frac { 1 } { 2 }\).
    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 $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 } { \sqrt { 1 - 4 x ^ { 2 } } }$$
OCR MEI Further Pure Core 2021 November Q2
4 marks Standard +0.3
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.
Edexcel CP2 2022 June Q5
6 marks Standard +0.3
  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.
Edexcel F3 2021 June Q4
8 marks Standard +0.8
  1. \(f(x) = x \arccos x \quad -1 \leq x \leq 1\) Find the exact value of \(f'(0.5)\). [3]
  2. \(g(x) = \arctan(e^{2x})\) Show that $$g''(x) = k \operatorname{sech}(2x) \tanh(2x)$$ where \(k\) is a constant to be found. [5]
Edexcel FP3 2011 June Q2
8 marks Standard +0.8
  1. Given that \(y = x \arcsin x\), \(0 \leq x \leq 1\), find
    1. an expression for \(\frac{dy}{dx}\),
    2. the exact value of \(\frac{dy}{dx}\) when \(x = \frac{1}{2}\).
    [3]
  2. Given that \(y = \arctan(3e^{2x})\), show that $$\frac{dy}{dx} = \frac{3}{5\cosh 2x + 4\sinh 2x}.$$ [5]
Edexcel FP3 Specimen Q5
7 marks Standard +0.8
Given that \(y = \arcsin x\) prove that
  1. \(\frac{dy}{dx} = \frac{1}{\sqrt{1-x^2}}\) [3]
  2. \((1-x^2) \frac{d^2 y}{dx^2} - x \frac{dy}{dx} = 0\) [4]
(Total 7 marks)
OCR FP2 2012 January Q6
8 marks Standard +0.8
  1. Prove that the derivative of \(\cos^{-1} x\) is \(-\frac{1}{\sqrt{1 - x^2}}\). [3]
A curve has equation \(y = \cos^{-1}(1 - x^2)\), for \(0 < x < \sqrt{2}\).
  1. Find and simplify \(\frac{dy}{dx}\), and hence show that $$(2 - x^2)\frac{d^2y}{dx^2} = x\frac{dy}{dx}.$$ [5]
SPS SPS FM Pure 2021 May Q7
9 marks Standard +0.3
Given that \(y = \arcsin x\), \(-1 \leqslant x < 1\),
  1. show that \(\frac{dy}{dx} = \frac{1}{\sqrt{1-x^2}}\). [3]
Given that \(f(x) = \frac{3x + 2}{\sqrt{4 - x^2}}\),
  1. show that the mean value of \(f(x)\) over the interval \([0, \sqrt{2}]\), is $$\frac{\pi\sqrt{2}}{4} + A\sqrt{2} - A,$$ where \(A\) is a constant to be determined. [6]