Find derivative of composite quotient/product

A question is this type if and only if it asks to differentiate a function involving both products and quotients, or products/quotients with chain rule, requiring multiple differentiation rules.

8 questions · Moderate -0.4

1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates
Sort by: Default | Easiest first | Hardest first
Edexcel C3 2010 January Q7
11 marks Standard +0.3
  1. (a) By writing \(\sec x\) as \(\frac { 1 } { \cos x }\), show that \(\frac { \mathrm { d } ( \sec x ) } { \mathrm { d } x } = \sec x \tan x\).
Given that \(y = \mathrm { e } ^ { 2 x } \sec 3 x\),
(b) find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\). The curve with equation \(y = \mathrm { e } ^ { 2 x } \sec 3 x , - \frac { \pi } { 6 } < x < \frac { \pi } { 6 }\), has a minimum turning point at \(( a , b )\).
(c) Find the values of the constants \(a\) and \(b\), giving your answers to 3 significant figures.
OCR MEI C3 2008 June Q8
18 marks Standard +0.3
8 Fig. 8 shows the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = \frac { 1 } { 1 + \cos x }\), for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\).
P is the point on the curve with \(x\)-coordinate \(\frac { 1 } { 3 } \pi\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8feffafd-4eba-4968-b4d2-88fa364d6170-3_825_816_571_662} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure}
  1. Find the \(y\)-coordinate of P .
  2. Find \(\mathrm { f } ^ { \prime } ( x )\). Hence find the gradient of the curve at the point P .
  3. Show that the derivative of \(\frac { \sin x } { 1 + \cos x }\) is \(\frac { 1 } { 1 + \cos x }\). Hence find the exact area of the region enclosed by the curve \(y = \mathrm { f } ( x )\), the \(x\)-axis, the \(y\)-axis and the line \(x = \frac { 1 } { 3 } \pi\).
  4. Show that \(\mathrm { f } ^ { - 1 } ( x ) = \arccos \left( \frac { 1 } { x } - 1 \right)\). State the domain of this inverse function, and add a sketch of \(y = \mathrm { f } ^ { - 1 } ( x )\) to a copy of Fig. 8.
AQA C3 2016 June Q1
6 marks Moderate -0.3
1
  1. Given that \(y = ( 4 x + 1 ) ^ { 3 } \sin 2 x\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Given that \(y = \frac { 2 x ^ { 2 } + 3 } { 3 x ^ { 2 } + 4 }\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { p x } { \left( 3 x ^ { 2 } + 4 \right) ^ { 2 } }\), where \(p\) is a constant.
  3. Given that \(y = \ln \left( \frac { 2 x ^ { 2 } + 3 } { 3 x ^ { 2 } + 4 } \right)\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
AQA C3 2011 June Q2
9 marks Moderate -0.3
    1. Find \(\frac{dy}{dx}\) when \(y = xe^{2x}\). [3]
    2. Find an equation of the tangent to the curve \(y = xe^{2x}\) at the point \((1, e^2)\). [2]
  2. Given that \(y = \frac{2\sin 3x}{1 + \cos 3x}\), use the quotient rule to show that $$\frac{dy}{dx} = \frac{k}{1 + \cos 3x}$$ where \(k\) is an integer. [4]
OCR C3 Q3
6 marks Moderate -0.3
  1. Differentiate \(x^2(x + 1)^6\) with respect to \(x\). [3]
  2. Find the gradient of the curve \(y = \frac{x^2 + 3}{x^2 - 3}\) at the point where \(x = 1\). [3]
OCR C3 2010 June Q1
6 marks Easy -1.2
Find \(\frac{dy}{dx}\) in each of the following cases:
  1. \(y = x^3 e^{2x}\), [2]
  2. \(y = \ln(3 + 2x^2)\), [2]
  3. \(y = \frac{x}{2x + 1}\). [2]
OCR C3 Q2
5 marks Moderate -0.8
Differentiate each of the following with respect to \(x\) and simplify your answers.
  1. \(\frac{6}{\sqrt{2x-7}}\) [2]
  2. \(x^2 e^{-x}\) [3]
SPS SPS SM Pure 2023 October Q1
8 marks Moderate -0.8
In all questions you must show all stages of your working, justifying solutions and not relying solely on calculator technology.
  1. Differentiate with respect to \(x\)
    1. \(x^2 e^{3x + 2}\), [4]
    2. \(\frac{\cos(2x^4)}{3x}\). [4]