4.08g Derivatives: inverse trig and hyperbolic functions

42 questions

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CAIE Further Paper 1 2024 November Q2
6 marks Challenging +1.2
Prove by mathematical induction that, for all positive integers \(n\), $$\frac{\mathrm{d}^n}{\mathrm{d}x^n}\left(\tan^{-1}x\right) = P_n(x)\left(1+x^2\right)^{-n},$$ where \(P_n(x)\) is a polynomial of degree \(n-1\). [6]
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 Q14
11 marks Challenging +1.2
The curve \(C\) has equation $$y = \operatorname{arcsec} e^x, \quad x > 0, \quad 0 < y < \frac{1}{2}\pi.$$
  1. Prove that \(\frac{dy}{dx} = \frac{1}{\sqrt{e^{2x} - 1}}\). [5]
  2. Sketch the graph of \(C\). [2]
The point \(A\) on \(C\) has \(x\)-coordinate \(\ln 2\). The tangent to \(C\) at \(A\) intersects the \(y\)-axis at the point \(B\).
  1. Find the exact value of the \(y\)-coordinate of \(B\). [4]
AQA FP2 2013 January Q5
11 marks Standard +0.8
  1. Using the definition \(\tanh y = \frac{\text{e}^y - \text{e}^{-y}}{\text{e}^y + \text{e}^{-y}}\), show that, for \(|x| < 1\), $$\tanh^{-1} x = \frac{1}{2} \ln \left(\frac{1+x}{1-x}\right)$$ [3 marks]
  2. Hence, or otherwise, show that \(\frac{\text{d}}{\text{d}x}(\tanh^{-1} x) = \frac{1}{1-x^2}\). [3 marks]
  3. Use integration by parts to show that $$\int_{0}^{\frac{1}{4}} \tanh^{-1} x \, \text{d}x = \ln \left(\frac{3^m}{2^n}\right)$$ where \(m\) and \(n\) are positive integers. [5 marks]
AQA Further Paper 1 2024 June Q11
5 marks Standard +0.3
  1. Find \(\frac{d}{dx}(x^2\tan^{-1} x)\) [1 mark]
  2. Hence find \(\int 2x \tan^{-1} x \, dx\) [4 marks]
AQA Further Paper 2 2020 June Q3
1 marks Moderate -0.5
Find the gradient of the tangent to the curve $$y = \sin^{-1} x$$ at the point where \(x = \frac{1}{5}\) Circle your answer. [1 mark] \(\frac{5\sqrt{6}}{12}\) \quad \(\frac{2\sqrt{6}}{5}\) \quad \(\frac{4\sqrt{3}}{25}\) \quad \(\frac{25}{24}\)
OCR MEI Further Pure Core Specimen Q12
13 marks Standard +0.3
In this question you must show detailed reasoning.
  1. Given that \(y = \arctan x\), show that \(\frac{dy}{dx} = \frac{1}{1+x^2}\). [3]
Fig. 12 shows the curve \(y = \frac{1}{1+x^2}\). \includegraphics{figure_12}
  1. Find, in exact form, the mean value of the function \(f(x) = \frac{1}{1+x^2}\) for \(-1 \leq x \leq 1\). [3]
  2. The region bounded by the curve, the \(x\)-axis, and the lines \(x = 1\) and \(x = -1\) is rotated through \(2\pi\) radians about the \(x\)-axis. Find, in exact form, the volume of the solid of revolution generated. [7]
WJEC Further Unit 4 2019 June Q4
16 marks Standard +0.3
  1. Given that \(y = \cot^{-1} x\), show that \(\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{-1}{x^2 + 1}\). [5]
  2. Express \(\frac{6x^2 - 10x - 9}{(2x + 3)(x^2 + 1)}\) in terms of partial fractions. [5]
  3. Hence find \(\int \frac{6x^2 - 8x - 6}{(2x + 3)(x^2 + 1)} \mathrm{d}x\). [5]
  4. Explain why \(\int_{-2}^{5} \frac{6x^2 - 8x - 6}{(2x + 3)(x^2 + 1)} \mathrm{d}x\) cannot be evaluated. [1]
WJEC Further Unit 4 2019 June Q9
14 marks Standard +0.8
  1. Given that \(y = \sin^{-1}(\cos \theta)\), where \(0 \leqslant \theta \leqslant \pi\), show that \(\frac{\mathrm{d}y}{\mathrm{d}\theta} = k\), where the value of \(k\) is to be determined. [4]
  2. Find the value of the gradient of the curve \(y = x^3 \tan^{-1} 4x\) when \(x = \frac{\pi}{2}\). [4]
  3. Find the equation of the normal to the curve \(y = \tanh^{-1}(1 - x)\) when \(x = 1.7\). [6]
WJEC Further Unit 4 2022 June Q11
15 marks Standard +0.8
  1. Differentiate each of the following with respect to \(x\).
    1. \(y = e^{3x}\sin^{-1}x\)
    2. \(y = \ln\left(\cosh^2(2x^2 + 7x)\right)\) [7]
  2. Find the equations of the tangents to the curve \(x = \sinh^{-1}(y^2)\) at the points where \(x = 1\). [8]
WJEC Further Unit 4 2023 June Q10
8 marks Standard +0.3
  1. By writing \(y = \sin^{-1}(2x + 5)\) as \(\sin y = 2x + 5\), show that \(\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{2}{\sqrt{1-(2x+5)^2}}\). [5]
  2. Deduce the range of values of \(x\) for which \(\frac{\mathrm{d}}{\mathrm{d}x}\left(\sin^{-1}(2x+5)\right)\) is valid. [3]
WJEC Further Unit 4 2024 June Q8
11 marks Challenging +1.2
  1. By writing \(y = \sinh^{-1}(4x + 3)\) as \(\sinh y = 4x + 3\), show that \(\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{4}{\sqrt{16x^2 + 24x + 10}}\). [5]
  2. Show that the graph of \(e^{-3x} \cdot y = \sinh 2x\) has only one stationary point. [6]
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]
Pre-U Pre-U 9795/1 2011 June Q13
18 marks Challenging +1.8
    1. Given that \(t = \tan x\), prove that \(\frac{2}{2 - \sin 2x} = \frac{1 + t^2}{1 - t + t^2}\). [2]
    2. Hence determine the value of the constant \(k\) for which $$\frac{d}{dx}\left\{\tan^{-1}\left(\frac{1 - 2\tan x}{\sqrt{3}}\right)\right\} = \frac{k}{2 - \sin 2x}.$$ [4]
  2. The curve \(C\) has cartesian equation \(x^2 - xy + y^2 = 72\).
    1. Determine a polar equation for \(C\) in the form \(r^2 = f(\theta)\), and deduce the polar coordinates \((r, \theta)\), where \(0 \leqslant \theta < 2\pi\), of the points on \(C\) which are furthest from the pole \(O\). [7]
    2. Find the exact area of the region of the plane in the first quadrant bounded by \(C\), the \(x\)-axis and the line \(y = x\). Deduce the total area of the region of the plane which lies inside \(C\) and within the first quadrant. [5]
Pre-U Pre-U 9795/1 2015 June Q13
10 marks Challenging +1.2
  1. By sketching a suitable triangle, show that \(\tan^{-1} a + \tan^{-1} \left(\frac{1}{a}\right) = \frac{1}{4}\pi\), for \(a > 0\). [1]
  2. Given that \(a\) and \(b\) are positive and less than 1, express \(\tan(\tan^{-1} a \pm \tan^{-1} b)\) in terms of \(a\) and \(b\). [2]
  3. By letting \(a = \frac{1}{n-1}\) and \(b = \frac{1}{n+1}\), use the method of differences to prove that $$\sum_{n=1}^{\infty} \tan^{-1} \left(\frac{2}{n^2}\right) = \frac{3}{4}\pi.$$ [7]
Pre-U Pre-U 9795 Specimen Q13
12 marks Standard +0.8
Given that \(y = \cos\{\ln(1 + x)\}\), prove that
  1. \((1 + x)\frac{\mathrm{d}y}{\mathrm{d}x} = -\sin\{\ln(1 + x)\}\), [1]
  2. \((1 + x)^2 \frac{\mathrm{d}^2 y}{\mathrm{d}x^2} + (1 + x)\frac{\mathrm{d}y}{\mathrm{d}x} + y = 0\). [2]
Obtain an equation relating \(\frac{\mathrm{d}^3 y}{\mathrm{d}x^3}\), \(\frac{\mathrm{d}^2 y}{\mathrm{d}x^2}\) and \(\frac{\mathrm{d}y}{\mathrm{d}x}\). [2] Hence find Maclaurin's series for \(y\), up to and including the term in \(x^3\). [4] Verify that the same result is obtained if the standard series expansions for \(\ln(1 + x)\) and \(\cos x\) are used. [3]