Basic power rule differentiation

Questions asking to differentiate simple polynomials and power functions (including negative and fractional powers) without composition, using only the power rule.

28 questions · Easy -1.1

1.07i Differentiate x^n: for rational n and sums
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CAIE P1 2003 June Q3
5 marks Easy -1.3
3
  1. Differentiate \(4 x + \frac { 6 } { x ^ { 2 } }\) with respect to \(x\).
  2. Find \(\int \left( 4 x + \frac { 6 } { x ^ { 2 } } \right) \mathrm { d } x\).
Edexcel C12 2018 October Q3
7 marks Easy -1.2
3. Given that \(y = 2 x ^ { 3 } - \frac { 5 } { 3 x ^ { 2 } } + 7 , x \neq 0\), find in its simplest form
  1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
  2. \(\int y \mathrm {~d} x\).
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Edexcel C1 2005 June Q2
5 marks Easy -1.2
Given that \(y = 6 x - \frac { 4 } { x ^ { 2 } } , x \neq 0\),
  1. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
  2. find \(\int y \mathrm {~d} x\).
OCR MEI C2 2005 June Q1
4 marks Easy -1.2
1 Differentiate \(x + \sqrt { x ^ { 3 } }\).
OCR MEI C2 Q4
5 marks Easy -1.8
4 Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when
  1. \(y = 2 x ^ { - 5 }\),
  2. \(y = \sqrt [ 3 ] { x }\).
OCR MEI C2 Q7
3 marks Easy -1.2
7 Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(y = \sqrt { x } + \frac { 3 } { x }\).
OCR MEI C2 2012 June Q1
3 marks Easy -1.2
1 Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(y = \sqrt { x } + \frac { 3 } { x }\).
OCR MEI C2 2015 June Q1
5 marks Easy -1.8
1
  1. Differentiate \(12 \sqrt [ 3 ] { x }\).
  2. Integrate \(\frac { 6 } { x ^ { 3 } }\).
OCR PURE Q1
6 marks Easy -1.2
1
  1. Find \(\frac { \mathrm { d } } { \mathrm { d } x } \left( x ^ { 3 } - 3 x + \frac { 5 } { x ^ { 2 } } \right)\).
  2. Find \(\int \left( 6 x ^ { 2 } - \frac { 2 } { x ^ { 3 } } \right) \mathrm { d } x\).
Edexcel C1 Q5
7 marks Easy -1.2
5. Given that $$y = x + 5 + \frac { 3 } { \sqrt { x } }$$
  1. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
  2. find \(\int y \mathrm {~d} x\).
Edexcel P1 2018 Specimen Q1
6 marks Easy -1.2
Given that \(y = 4x^3 - \frac{5}{x^2}\), \(x \neq 0\), find in their simplest form
  1. \(\frac{dy}{dx}\). [3]
  2. \(\int y \, dx\) [3]
Edexcel C1 Q2
8 marks Easy -1.8
  1. Given that \(y = 5x^3 + 7x + 3\), find
    1. \(\frac{dy}{dx}\), [3]
    2. \(\frac{d^2y}{dx^2}\). [1]
  2. Find \(\int \left(1 + 3\sqrt{x} - \frac{1}{x^2}\right) dx\). [4]
Edexcel C1 Q2
5 marks Easy -1.2
Given that \(y = 6x - \frac{4}{x^2}\), \(x \neq 0\),
  1. find \(\frac{dy}{dx}\), [2]
  2. find \(\int y \, dx\). [3]
Edexcel C1 Q4
5 marks Easy -1.2
Given that \(y = 2x^2 - \frac{6}{x}\), \(x \neq 0\),
  1. find \(\frac{dy}{dx}\), [2]
  2. find \(\int y \, dx\). [3]
Edexcel C1 Q43
5 marks Moderate -0.3
\(f(x) = \frac{(x^2 - 3)^2}{x^3}, x \neq 0\).
  1. Show that \(f(x) \equiv x - 6x^{-1} + 9x^{-3}\). [2]
  2. Hence, or otherwise, differentiate \(f(x)\) with respect to \(x\). [3]
Edexcel C2 Q31
10 marks Moderate -0.8
  1. Differentiate with respect to \(x\) $$2x^3 + \sqrt{x} + \frac{x^2 + 2x}{x^2}.$$ [5]
  2. Evaluate $$\int_1^4 \left(\frac{x}{2} + \frac{1}{x^2}\right) dx.$$ [5]
Edexcel C1 Q3
5 marks Moderate -0.3
Differentiate with respect to \(x\) $$\frac{6x^2 - 1}{2\sqrt{x}}.$$ [5]
Edexcel C1 Q5
6 marks Moderate -0.3
Given that $$y = \frac{x^4 - 3}{2x^2},$$
  1. find \(\frac{dy}{dx}\), [4]
  2. show that \(\frac{d^2y}{dx^2} = \frac{x^4 - 9}{x^4}\). [2]
Edexcel C1 Q7
8 marks Moderate -0.8
Given that $$y = \sqrt{x} - \frac{4}{\sqrt{x}},$$
  1. find \(\frac{dy}{dx}\). [3]
  2. find \(\frac{d^2y}{dx^2}\). [2]
  3. show that $$4x\frac{d^2y}{dx^2} + 4x\frac{dy}{dx} - y = 0.$$ [3]
Edexcel C2 Q2
10 marks Moderate -0.8
  1. Differentiate with respect to x $$2x^3 + \sqrt{x} + \frac{x^2 + 2x}{x^2}.$$ [5 marks]
  2. Evaluate $$\int_1^4 \left(\frac{x}{2} + \frac{1}{x^2}\right) dx.$$ [5 marks]
OCR MEI C2 2006 June Q8
5 marks Easy -1.2
Given that \(y = 6x^3 + \sqrt{x} + 3\), find \(\frac{dy}{dx}\) and \(\frac{d^2y}{dx^2}\). [5]
OCR MEI C2 Q1
4 marks Easy -1.2
Differentiate \(x + \sqrt{x^3}\). [4]
OCR MEI C2 Q6
2 marks Easy -1.8
Differentiate \(10x^4 + 12\). [2]
OCR MEI C2 Q8
2 marks Easy -1.2
Differentiate \(6x^{\frac{5}{2}} + 4\). [2]
Edexcel C3 Q1
Standard +0.0
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