Chain rule with single composition

Questions asking to differentiate composite functions of the form f(g(x)) where the chain rule is required, such as (ax+b)^n, √(ax²+b), or similar single-layer compositions.

10 questions · Moderate -0.7

1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates
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OCR MEI C3 2008 January Q1
4 marks Moderate -0.8
1 Differentiate \(\sqrt [ 3 ] { 1 + 6 x ^ { 2 } }\).
OCR MEI C3 Q2
3 marks Moderate -0.8
2 Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(y = \sqrt { 1 + x ^ { 3 } }\).
OCR MEI C3 Q3
6 marks Moderate -0.8
3 Differentiate the following functions.
  1. \(\quad y = \left( x ^ { 2 } + 3 \right) ^ { 5 }\)
  2. \(y = \frac { \sin 2 x } { x }\)
OCR MEI C3 Q6
4 marks Moderate -0.8
6 Given that \(y = \sqrt [ 3 ] { 1 + x ^ { 2 } }\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
WJEC Unit 3 2022 June Q2
Moderate -0.8
Differentiate the following functions with respect to \(x\). a) \(x ^ { 3 } \ln ( 5 x )\) b) \(( x + \cos 3 x ) ^ { 4 }\)
CAIE P1 2023 June Q11
8 marks Moderate -0.3
The equation of a curve is $$y = k\sqrt{4x + 1} - x + 5,$$ where \(k\) is a positive constant.
  1. Find \(\frac{dy}{dx}\). [2]
  2. Find the \(x\)-coordinate of the stationary point in terms of \(k\). [2]
  3. Given that \(k = 10.5\), find the equation of the normal to the curve at the point where the tangent to the curve makes an angle of \(\tan^{-1}(2)\) with the positive \(x\)-axis. [4]
OCR MEI C3 2011 January Q1
7 marks Moderate -0.8
Given that \(y = \sqrt[3]{1 + x^2}\), find \(\frac{dy}{dx}\). [4]
OCR MEI C3 Q2
4 marks Moderate -0.3
Differentiate \(\sqrt{1 + 6x^2}\). [4]
AQA Paper 1 2024 June Q6
2 marks Easy -1.2
Use the chain rule to find \(\frac{dy}{dx}\) when \(y = (x^3 + 5x)^7\) [2 marks]
SPS SPS FM 2021 March Q1
10 marks Moderate -0.8
Differentiate the following with respect to \(x\), simplifying your answers fully
  1. \(y = e^{3x} + \ln 2x\) [1]
  2. \(y = (5 + x^2)^{\frac{3}{2}}\) [2]
  3. \(y = \frac{2x}{(5-3x^2)^{\frac{1}{2}}}\) [4]
  4. \(y = e^{-\frac{3}{x}} \ln(1 + x^3)\) [3]