1.07s Parametric and implicit differentiation

761 questions

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Edexcel C3 2006 January Q4
13 marks Moderate -0.3
4. (a) Differentiate with respect to \(x\)
  1. \(x ^ { 2 } \mathrm { e } ^ { 3 x + 2 }\),
  2. \(\frac { \cos \left( 2 x ^ { 3 } \right) } { 3 x }\).
    (b) Given that \(x = 4 \sin ( 2 y + 6 )\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\).
Edexcel C3 2007 January Q3
9 marks Moderate -0.3
3. The curve \(C\) has equation $$x = 2 \sin y .$$
  1. Show that the point \(P \left( \sqrt { } 2 , \frac { \pi } { 4 } \right)\) lies on \(C\).
  2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \sqrt { 2 } }\) at \(P\).
  3. Find an equation of the normal to \(C\) at \(P\). Give your answer in the form \(y = m x + c\), where \(m\) and \(c\) are exact constants.
Edexcel C3 2009 January Q4
6 marks Standard +0.3
4. Find the equation of the tangent to the curve \(x = \cos ( 2 y + \pi )\) at \(\left( 0 , \frac { \pi } { 4 } \right)\). Give your answer in the form \(y = a x + b\), where \(a\) and \(b\) are constants to be found.
Edexcel C3 2010 January Q4
9 marks Moderate -0.3
4.
  1. Given that \(y = \frac { \ln \left( x ^ { 2 } + 1 \right) } { x }\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Given that \(x = \tan y\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 1 + x ^ { 2 } }\).
Edexcel C3 2011 January Q8
9 marks Moderate -0.3
8.
  1. 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 $$x = \sec 2 y$$
  2. find \(\frac { \mathrm { d } x } { \mathrm {~d} y }\) in terms of \(y\).
  3. 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 C3 2012 January Q4
7 marks Standard +0.3
4. The point \(P\) is the point on the curve \(x = 2 \tan \left( y + \frac { \pi } { 12 } \right)\) with \(y\)-coordinate \(\frac { \pi } { 4 }\). Find an equation of the normal to the curve at \(P\).
Edexcel C3 2013 January Q5
11 marks Moderate -0.3
5. (i) Differentiate with respect to \(x\)
  1. \(y = x ^ { 3 } \ln 2 x\)
  2. \(y = ( x + \sin 2 x ) ^ { 3 }\) Given that \(x = \cot y\),
    (ii) show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { - 1 } { 1 + x ^ { 2 } }\)
Edexcel C3 2014 January Q5
9 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6f22eb1b-21de-45f1-9a8a-deac7ac8d0b0-14_646_1013_207_532} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The curve shown in Figure 1 has equation $$x = 3 \sin y + 3 \cos y , \quad - \frac { \pi } { 4 } < y < \frac { \pi } { 4 }$$
  1. Express the equation of the curve in the form \(x = R \sin ( y + \alpha )\), where \(R\) and \(\alpha\) are constants, \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\)
  2. Find the coordinates of the point on the curve where the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) is \(\frac { 1 } { 2 }\). Give your answers to 3 decimal places.
Edexcel C3 2012 June Q7
11 marks Moderate -0.3
  1. Differentiate with respect to \(x\),
    1. \(x ^ { \frac { 1 } { 2 } } \ln ( 3 x )\)
    2. \(\frac { 1 - 10 x } { ( 2 x - 1 ) ^ { 5 } }\), giving your answer in its simplest form.
  2. Given that \(x = 3 \tan 2 y\) find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\).
Edexcel C3 2013 June Q5
10 marks Moderate -0.3
5.
  1. Differentiate $$\frac { \cos 2 x } { \sqrt { x } }$$ with respect to \(x\).
  2. Show that \(\frac { \mathrm { d } } { \mathrm { d } x } \left( \sec ^ { 2 } 3 x \right)\) can be written in the form $$\mu \left( \tan 3 x + \tan ^ { 3 } 3 x \right)$$ where \(\mu\) is a constant.
  3. Given \(x = 2 \sin \left( \frac { y } { 3 } \right)\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\), simplifying your answer.
Edexcel C3 2013 June Q9
8 marks Standard +0.3
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0f6fd881-4d4b-4f80-96cc-6da41cc33c60-16_570_903_237_534} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 shows a sketch of the curve with equation \(x = \left( 9 + 16 y - 2 y ^ { 2 } \right) ^ { \frac { 1 } { 2 } }\).
The curve crosses the \(x\)-axis at the point \(A\).
  1. State the coordinates of \(A\).
  2. Find an expression for \(\frac { \mathrm { d } x } { \mathrm {~d} y }\), in terms of \(y\).
  3. Find an equation of the tangent to the curve at \(A\).
Edexcel C3 2013 June Q5
10 marks Standard +0.3
  1. Given that
$$x = \sec ^ { 2 } 3 y , \quad 0 < y < \frac { \pi } { 6 }$$
  1. find \(\frac { \mathrm { d } x } { \mathrm {~d} y }\) in terms of \(y\).
  2. Hence show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 6 x ( x - 1 ) ^ { \frac { 1 } { 2 } } }$$
  3. Find an expression for \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) in terms of \(x\). Give your answer in its simplest form.
Edexcel C3 2014 June Q4
12 marks Moderate -0.3
  1. Given that $$x = \sec ^ { 2 } 2 y , \quad 0 < y < \frac { \pi } { 4 }$$ show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 4 x \sqrt { ( x - 1 ) } }$$
  2. Given that $$y = \left( x ^ { 2 } + x ^ { 3 } \right) \ln 2 x$$ find the exact value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at \(x = \frac { \mathrm { e } } { 2 }\), giving your answer in its simplest form.
  3. Given that $$f ( x ) = \frac { 3 \cos x } { ( x + 1 ) ^ { \frac { 1 } { 3 } } } , \quad x \neq - 1$$ show that $$\mathrm { f } ^ { \prime } ( x ) = \frac { \mathrm { g } ( x ) } { ( x + 1 ) ^ { \frac { 4 } { 3 } } } , \quad x \neq - 1$$ where \(\mathrm { g } ( x )\) is an expression to be found.
Edexcel C3 2014 June Q3
8 marks Standard +0.3
3. The curve \(C\) has equation \(x = 8 y \tan 2 y\) The point \(P\) has coordinates \(\left( \pi , \frac { \pi } { 8 } \right)\)
  1. Verify that \(P\) lies on \(C\).
  2. Find the equation of the tangent to \(C\) at \(P\) in the form \(a y = x + b\), where the constants \(a\) and \(b\) are to be found in terms of \(\pi\).
Edexcel C3 2015 June Q5
7 marks Standard +0.3
5. The point \(P\) lies on the curve with equation $$x = ( 4 y - \sin 2 y ) ^ { 2 }$$ Given that \(P\) has \(( x , y )\) coordinates \(\left( p , \frac { \pi } { 2 } \right)\), where \(p\) is a constant,
  1. find the exact value of \(p\). The tangent to the curve at \(P\) cuts the \(y\)-axis at the point \(A\).
  2. Use calculus to find the coordinates of \(A\).
Edexcel C3 2016 June Q5
10 marks Standard +0.3
5.
  1. Find, using calculus, the \(x\) coordinate of the turning point of the curve with equation $$y = \mathrm { e } ^ { 3 x } \cos 4 x , \quad \frac { \pi } { 4 } \leqslant x < \frac { \pi } { 2 }$$ Give your answer to 4 decimal places.
  2. Given \(x = \sin ^ { 2 } 2 y , \quad 0 < y < \frac { \pi } { 4 }\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) as a function of \(y\). Write your answer in the form $$\frac { \mathrm { d } y } { \mathrm {~d} x } = p \operatorname { cosec } ( q y ) , \quad 0 < y < \frac { \pi } { 4 }$$ where \(p\) and \(q\) are constants to be determined. \includegraphics[max width=\textwidth, alt={}, center]{d3ba2776-eedb-48f0-834f-41aa454afba3-09_2258_47_315_37}
Edexcel C3 2017 June Q7
10 marks Standard +0.3
  1. Given \(y = 2 x \left( x ^ { 2 } - 1 \right) ^ { 5 }\), show that
    1. \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { g } ( x ) \left( x ^ { 2 } - 1 \right) ^ { 4 }\) where \(\mathrm { g } ( x )\) is a function to be determined.
    2. Hence find the set of values of \(x\) for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } \geqslant 0\)
    3. Given $$x = \ln ( \sec 2 y ) , \quad 0 < y < \frac { \pi } { 4 }$$ find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) as a function of \(x\) in its simplest form.
Edexcel C3 2018 June Q8
7 marks Standard +0.3
  1. By writing \(\sec \theta = \frac { 1 } { \cos \theta }\), show that \(\frac { \mathrm { d } } { \mathrm { d } \theta } ( \sec \theta ) = \sec \theta \tan \theta\)
  2. Given that $$x = \mathrm { e } ^ { \sec y } \quad x > \mathrm { e } , \quad 0 < y < \frac { \pi } { 2 }$$ show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { x \sqrt { \mathrm {~g} ( x ) } } , \quad x > \mathrm { e }$$ where \(\mathrm { g } ( x )\) is a function of \(\ln x\).
Edexcel C3 Q4
12 marks Moderate -0.3
4. Differentiate with respect to \(x\)
  1. \(x ^ { 3 } \mathrm { e } ^ { 3 x }\),
  2. \(\frac { 2 x } { \cos x }\),
  3. \(\tan ^ { 2 } x\). Given that \(x = \cos y ^ { 2 }\),
  4. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(y\).
Edexcel C3 Specimen Q7
14 marks Moderate -0.3
7.
  1. Given that \(y = \tan x + 2 \cos x\), find the exact value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at \(x = \frac { \pi } { 4 }\).
  2. Given that \(x = \tan \frac { 1 } { 2 } y\), prove that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 } { 1 + x ^ { 2 } }\).
  3. Given that \(y = \mathrm { e } ^ { - x } \sin 2 x\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) can be expressed in the form \(R \mathrm { e } ^ { - x } \cos ( 2 x + \alpha )\). Find, to 3 significant figures, the values of \(R\) and \(\alpha\), where \(0 < \alpha < \frac { \pi } { 2 }\).
Edexcel P4 2021 January Q6
9 marks Standard +0.8
6. A curve has equation $$4 y ^ { 2 } + 3 x = 6 y \mathrm { e } ^ { - 2 x }$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\). The curve crosses the \(y\)-axis at the origin and at the point \(P\).
  2. Find the equation of the normal to the curve at \(P\), writing your answer in the form \(y = m x + c\) where \(m\) and \(c\) are constants to be found.
Edexcel P4 2022 January Q3
10 marks Standard +0.3
3. The curve \(C\) has equation $$3 y ^ { 2 } - 11 x ^ { 2 } + 11 x y = 20 y - 36 x + 28$$
  1. Find, in simplest form, \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\). The point \(P ( 4 , k )\), where \(k\) is a constant, lies on \(C\).
    Given that \(k < 0\)
  2. find the value of the gradient of \(C\) at \(P\)
Edexcel P4 2022 January Q1
6 marks Standard +0.3
  1. The curve \(C\) has equation
$$x y ^ { 2 } = x ^ { 2 } y + 6 \quad x \neq 0 \quad y \neq 0$$ Find an equation for the tangent to \(C\) at the point \(P ( 2,3 )\), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.
(6)
Edexcel P4 2023 January Q5
7 marks Challenging +1.2
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c46ca445-cf59-4664-931e-add9f2f81851-14_940_881_251_593} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of the curve with equation $$y ^ { 2 } = 2 x ^ { 2 } + 15 x + 10 y$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\). The curve is not defined for values of \(x\) in the interval ( \(p , q\) ), as shown in Figure 2.
  2. Using your answer to part (a) or otherwise, find the value of \(p\) and the value of \(q\).
    (Solutions relying entirely on calculator technology are not acceptable.)
Edexcel P4 2023 January Q8
11 marks Challenging +1.2
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c46ca445-cf59-4664-931e-add9f2f81851-26_582_773_255_648} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} \section*{In this question you must show all stages of your working.} \section*{Solutions relying entirely on calculator technology are not acceptable.} A curve \(C\) has parametric equations $$x = \sin ^ { 2 } t \quad y = 2 \tan t \quad 0 \leqslant t < \frac { \pi } { 2 }$$ The point \(P\) with parameter \(t = \frac { \pi } { 4 }\) lies on \(C\).
The line \(l\) is the normal to \(C\) at \(P\), as shown in Figure 3.
  1. Show, using calculus, that an equation for \(l\) is $$8 y + 2 x = 17$$ The region \(S\), shown shaded in Figure 3, is bounded by \(C , l\) and the \(x\)-axis.
  2. Find, using calculus, the exact area of \(S\).