Inverse function differentiation

Questions involving differentiation of inverse trigonometric functions or finding dy/dx when x is given as a function of y, using dx/dy and the chain rule.

5 questions · Standard +0.3

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Edexcel C34 2018 January Q8
4 marks Standard +0.3
  1. Given that
$$y = 8 \tan ( 2 x ) , \quad - \frac { \pi } { 4 } < x < \frac { \pi } { 4 }$$ show that $$\frac { \mathrm { d } x } { \mathrm {~d} y } = \frac { A } { B + y ^ { 2 } }$$ where \(A\) and \(B\) are integers to be found.
Edexcel C3 2011 January Q8
9 marks Moderate -0.3
8. (a) Given that $$\frac { \mathrm { d } } { \mathrm {~d} x } ( \cos x ) = - \sin x$$ show that \(\frac { \mathrm { d } } { \mathrm { d } x } ( \sec x ) = \sec x \tan x\). Given that $$x = \sec 2 y$$ (b) find \(\frac { \mathrm { d } x } { \mathrm {~d} y }\) in terms of \(y\).
(c) Hence find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\). \includegraphics[max width=\textwidth, alt={}, center]{3ff6824f-9fbf-4b5b-8bab-91332c549b36-14_102_93_2473_1804}
Edexcel F3 2014 June Q1
6 marks Standard +0.8
  1. Given that \(y = \arctan \left( \frac { 2 x } { 3 } \right)\),
    1. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), giving your answer in its simplest form.
    2. Use integration by parts to find
    $$\int \arctan \left( \frac { 2 x } { 3 } \right) \mathrm { d } x$$
OCR MEI C3 Q7
7 marks Standard +0.3
Fig. 3 shows the curve defined by the equation \(y = \arcsin(x - 1)\), for \(0 \leqslant x \leqslant 2\). \includegraphics{figure_7}
  1. Find \(x\) in terms of \(y\), and show that \(\frac{dx}{dy} = \cos y\). [3]
  2. Hence find the exact gradient of the curve at the point where \(x = 1.5\). [4]
OCR H240/01 2017 Specimen Q10
8 marks Standard +0.3
A curve has equation \(x = (y + 5)\ln(2y - 7)\).
  1. Find \(\frac{dx}{dy}\) in terms of y. [3]
  2. Find the gradient of the curve where it crosses the y-axis. [5]