Show dy/dx simplifies to given form

A question is this type if and only if it asks the student to derive and show that dy/dx equals a specific given expression (e.g. cot θ, tan(t − π/4)), requiring algebraic or trigonometric simplification as the main task.

16 questions · Standard +0.0

1.07s Parametric and implicit differentiation
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CAIE P2 2003 June Q7
11 marks Standard +0.3
7 The parametric equations of a curve are $$x = 2 \theta - \sin 2 \theta , \quad y = 2 - \cos 2 \theta$$
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \cot \theta\).
  2. Find the equation of the tangent to the curve at the point where \(\theta = \frac { 1 } { 4 } \pi\).
  3. For the part of the curve where \(0 < \theta < 2 \pi\), find the coordinates of the points where the tangent is parallel to the \(x\)-axis.
CAIE P2 2004 June Q6
10 marks Moderate -0.3
6 The parametric equations of a curve are $$x = 2 t + \ln t , \quad y = t + \frac { 4 } { t }$$ where \(t\) takes all positive values.
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { t ^ { 2 } - 4 } { t ( 2 t + 1 ) }\).
  2. Find the equation of the tangent to the curve at the point where \(t = 1\).
  3. The curve has one stationary point. Find the \(y\)-coordinate of this point, and determine whether this point is a maximum or a minimum.
CAIE P3 2012 June Q3
5 marks Moderate -0.3
3 The parametric equations of a curve are $$x = \sin 2 \theta - \theta , \quad y = \cos 2 \theta + 2 \sin \theta$$ Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 \cos \theta } { 1 + 2 \sin \theta }\).
CAIE P3 2008 November Q4
5 marks Moderate -0.5
4 The parametric equations of a curve are $$x = a ( 2 \theta - \sin 2 \theta ) , \quad y = a ( 1 - \cos 2 \theta )$$ Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \cot \theta\).
CAIE P3 2013 November Q4
6 marks Standard +0.3
4 The parametric equations of a curve are $$x = \mathrm { e } ^ { - t } \cos t , \quad y = \mathrm { e } ^ { - t } \sin t$$ Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \tan \left( t - \frac { 1 } { 4 } \pi \right)\).
CAIE P3 2014 November Q4
7 marks Standard +0.3
4 The parametric equations of a curve are $$x = \frac { 1 } { \cos ^ { 3 } t } , \quad y = \tan ^ { 3 } t$$ where \(0 \leqslant t < \frac { 1 } { 2 } \pi\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \sin t\).
  2. Hence show that the equation of the tangent to the curve at the point with parameter \(t\) is \(y = x \sin t - \tan t\).
CAIE P3 2014 November Q2
5 marks Standard +0.3
2 A curve is defined for \(0 < \theta < \frac { 1 } { 2 } \pi\) by the parametric equations $$x = \tan \theta , \quad y = 2 \cos ^ { 2 } \theta \sin \theta$$ Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 \cos ^ { 5 } \theta - 4 \cos ^ { 3 } \theta\).
CAIE P3 2019 November Q3
5 marks Moderate -0.3
3 The parametric equations of a curve are $$x = 2 t + \sin 2 t , \quad y = \ln ( 1 - \cos 2 t )$$ Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \operatorname { cosec } 2 t\).
CAIE P3 2023 June Q4
5 marks Standard +0.3
4 The parametric equations of a curve are $$x = \frac { \cos \theta } { 2 - \sin \theta } , \quad y = \theta + 2 \cos \theta$$ Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = ( 2 - \sin \theta ) ^ { 2 }\).
CAIE P3 2022 March Q4
5 marks Moderate -0.3
4 The parametric equations of a curve are $$x = 1 - \cos \theta , \quad y = \cos \theta - \frac { 1 } { 4 } \cos 2 \theta$$ Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - 2 \sin ^ { 2 } \left( \frac { 1 } { 2 } \theta \right)\).
CAIE P3 2020 November Q3
5 marks Moderate -0.8
3 The parametric equations of a curve are $$x = 3 - \cos 2 \theta , \quad y = 2 \theta + \sin 2 \theta$$ for \(0 < \theta < \frac { 1 } { 2 } \pi\).
Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \cot \theta\).
CAIE P3 2022 November Q4
5 marks Standard +0.3
4 The parametric equations of a curve are $$x = 2 t - \tan t , \quad y = \ln ( \sin 2 t )$$ for \(0 < t < \frac { 1 } { 2 } \pi\).
Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \cot t\).
Pre-U Pre-U 9794/1 2015 June Q10
11 marks Standard +0.3
10 A curve has parametric equations given by $$x = - \sqrt { ( 1 - t ) ^ { 3 } } \quad y = \sqrt { ( 1 + t ) ^ { 3 } } \quad \text { for } - 1 < t < 1$$
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 + t } { \sqrt { 1 - t ^ { 2 } } }\).
  2. Write \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) as a series of ascending powers of \(t\) up to and including the term in \(t ^ { 4 }\), and hence estimate the gradient of the curve when \(t = 0.5\).
CAIE P3 2006 June Q3
5 marks Moderate -0.3
The parametric equations of a curve are $$x = 2\theta + \sin 2\theta, \quad y = 1 - \cos 2\theta.$$ Show that \(\frac{dy}{dx} = \tan \theta\). [5]
CAIE P3 2013 November Q4
6 marks Standard +0.3
The parametric equations of a curve are $$x = e^{-t}\cos t, \quad y = e^{-t}\sin t.$$ Show that \(\frac{dy}{dx} = \tan(t - \frac{1}{4}\pi)\). [6]
SPS SPS FM Pure 2023 June Q11
7 marks Challenging +1.2
In this question you must show detailed reasoning. A curve has parametric equations $$x = \cos t - 3t \text{ and } y = 3t - 4\cos t - \sin 2t, \text{ for } 0 \leqslant t \leqslant \pi.$$ Show that the gradient of the curve is always negative. [7]