1.07i Differentiate x^n: for rational n and sums

726 questions

Sort by: Default | Easiest first | Hardest first
Edexcel C1 2009 January Q6
6 marks Easy -1.3
Given that \(\frac { 2 x ^ { 2 } - x ^ { \frac { 3 } { 2 } } } { \sqrt { } x }\) can be written in the form \(2 x ^ { p } - x ^ { q }\),
  1. write down the value of \(p\) and the value of \(q\). Given that \(y = 5 x ^ { 4 } - 3 + \frac { 2 x ^ { 2 } - x ^ { \frac { 3 } { 2 } } } { \sqrt { } x }\),
  2. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), simplifying the coefficient of each term.
Edexcel C1 2009 January Q11
13 marks Moderate -0.3
  1. The curve \(C\) has equation
$$y = 9 - 4 x - \frac { 8 } { x } , \quad x > 0$$ The point \(P\) on \(C\) has \(x\)-coordinate equal to 2 .
  1. Show that the equation of the tangent to \(C\) at the point \(P\) is \(y = 1 - 2 x\).
  2. Find an equation of the normal to \(C\) at the point \(P\). The tangent at \(P\) meets the \(x\)-axis at \(A\) and the normal at \(P\) meets the \(x\)-axis at \(B\).
  3. Find the area of triangle \(A P B\).
Edexcel C1 2010 January Q1
3 marks Easy -1.8
Given that \(y = x ^ { 4 } + x ^ { \frac { 1 } { 3 } } + 3\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
Edexcel C1 2010 January Q6
8 marks Moderate -0.8
6. The curve \(C\) has equation $$y = \frac { ( x + 3 ) ( x - 8 ) } { x } , \quad x > 0$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in its simplest form.
  2. Find an equation of the tangent to \(C\) at the point where \(x = 2\)
Edexcel C1 2011 January Q11
12 marks Moderate -0.8
11. The curve \(C\) has equation $$y = \frac { 1 } { 2 } x ^ { 3 } - 9 x ^ { \frac { 3 } { 2 } } + \frac { 8 } { x } + 30 , \quad x > 0$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Show that the point \(P ( 4 , - 8 )\) lies on \(C\).
  3. Find an equation of the normal to \(C\) at the point \(P\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers. \includegraphics[max width=\textwidth, alt={}, center]{95e11fd7-765c-477d-800b-7574bc1af81f-15_113_129_2405_1816}
Edexcel C1 2012 January Q1
6 marks Easy -1.2
Given that \(y = x ^ { 4 } + 6 x ^ { \frac { 1 } { 2 } }\), find in their simplest form
  1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
  2. \(\int y \mathrm {~d} x\)
Edexcel C1 2012 January Q8
10 marks Moderate -0.8
8. The curve \(C _ { 1 }\) has equation $$y = x ^ { 2 } ( x + 2 )$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
  2. Sketch \(C _ { 1 }\), showing the coordinates of the points where \(C _ { 1 }\) meets the \(x\)-axis.
  3. Find the gradient of \(C _ { 1 }\) at each point where \(C _ { 1 }\) meets the \(x\)-axis. The curve \(C _ { 2 }\) has equation $$y = ( x - k ) ^ { 2 } ( x - k + 2 )$$ where \(k\) is a constant and \(k > 2\)
  4. Sketch \(C _ { 2 }\), showing the coordinates of the points where \(C _ { 2 }\) meets the \(x\) and \(y\) axes.
Edexcel C1 2013 January Q11
12 marks Moderate -0.8
11. The curve \(C\) has equation $$y = 2 x - 8 \sqrt { } x + 5 , \quad x \geqslant 0$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), giving each term in its simplest form. The point \(P\) on \(C\) has \(x\)-coordinate equal to \(\frac { 1 } { 4 }\)
  2. Find the equation of the tangent to \(C\) at the point \(P\), giving your answer in the form \(y = a x + b\), where \(a\) and \(b\) are constants. The tangent to \(C\) at the point \(Q\) is parallel to the line with equation \(2 x - 3 y + 18 = 0\)
  3. Find the coordinates of \(Q\).
Edexcel C1 2014 January Q2
5 marks Easy -1.3
2. $$y = 2 x ^ { 2 } - \frac { 4 } { \sqrt { } x } + 1 , \quad x > 0$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), giving each term in its simplest form.
  2. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\), giving each term in its simplest form. \includegraphics[max width=\textwidth, alt={}, center]{6081d81b-51d2-4140-9834-71ef7fd700b0-05_104_97_2613_1784}
Edexcel C1 2014 January Q10
10 marks Moderate -0.3
10. The curve \(C\) has equation \(y = x ^ { 3 } - 2 x ^ { 2 } - x + 3\) The point \(P\), which lies on \(C\), has coordinates \(( 2,1 )\).
  1. Show that an equation of the tangent to \(C\) at the point \(P\) is \(y = 3 x - 5\) The point \(Q\) also lies on \(C\).
    Given that the tangent to \(C\) at \(Q\) is parallel to the tangent to \(C\) at \(P\),
  2. find the coordinates of the point \(Q\).
Edexcel C1 2005 June Q10
11 marks Moderate -0.3
10. The curve \(C\) has equation \(y = \frac { 1 } { 3 } x ^ { 3 } - 4 x ^ { 2 } + 8 x + 3\). The point \(P\) has coordinates \(( 3,0 )\).
  1. Show that \(P\) lies on \(C\).
  2. Find the equation of the tangent to \(C\) at \(P\), giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants. Another point \(Q\) also lies on \(C\). The tangent to \(C\) at \(Q\) is parallel to the tangent to \(C\) at \(P\).
  3. Find the coordinates of \(Q\).
Edexcel C1 2006 June Q5
7 marks Easy -1.3
5. Differentiate with respect to \(x\)
  1. \(x ^ { 4 } + 6 \sqrt { } x\),
  2. \(\frac { ( x + 4 ) ^ { 2 } } { x }\).
Edexcel C1 2007 June Q10
13 marks Moderate -0.3
10. The curve \(C\) has equation \(y = x ^ { 2 } ( x - 6 ) + \frac { 4 } { x } , x > 0\). The points \(P\) and \(Q\) lie on \(C\) and have \(x\)-coordinates 1 and 2 respectively.
  1. Show that the length of \(P Q\) is \(\sqrt { 170 }\).
  2. Show that the tangents to \(C\) at \(P\) and \(Q\) are parallel.
  3. Find an equation for the normal to \(C\) at \(P\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers. \(\_\_\_\_\)
Edexcel C1 2008 June Q4
5 marks Easy -1.3
4. $$\mathrm { f } ( x ) = 3 x + x ^ { 3 } , \quad x > 0 .$$
  1. Differentiate to find \(\mathrm { f } ^ { \prime } ( x )\). Given that \(\mathrm { f } ^ { \prime } ( x ) = 15\),
  2. find the value of \(x\).
Edexcel C1 2008 June Q11
8 marks Moderate -0.8
The gradient of a curve \(C\) is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { \left( x ^ { 2 } + 3 \right) ^ { 2 } } { x ^ { 2 } } , x \neq 0\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = x ^ { 2 } + 6 + 9 x ^ { - 2 }\). The point \(( 3,20 )\) lies on \(C\).
  2. Find an equation for the curve \(C\) in the form \(y = \mathrm { f } ( x )\).
Edexcel C1 2009 June Q3
6 marks Easy -1.2
3. Given that \(y = 2 x ^ { 3 } + \frac { 3 } { x ^ { 2 } } , x \neq 0\), find
  1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
  2. \(\int y \mathrm {~d} x\), simplifying each term.
Edexcel C1 2009 June Q9
8 marks Moderate -0.8
9. $$f ( x ) = \frac { ( 3 - 4 \sqrt { } x ) ^ { 2 } } { \sqrt { } x } , \quad x > 0$$
  1. Show that \(\mathrm { f } ( x ) = 9 x ^ { - \frac { 1 } { 2 } } + A x ^ { \frac { 1 } { 2 } } + B\), where \(A\) and \(B\) are constants to be found.
  2. Find \(\mathrm { f } ^ { \prime } ( x )\).
  3. Evaluate \(\mathrm { f } ^ { \prime } ( 9 )\).
Edexcel C1 2009 June Q11
11 marks Standard +0.8
11. The curve \(C\) has equation $$y = x ^ { 3 } - 2 x ^ { 2 } - x + 9 , \quad x > 0$$ The point \(P\) has coordinates (2, 7).
  1. Show that \(P\) lies on \(C\).
  2. Find the equation of the tangent to \(C\) at \(P\), giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants. The point \(Q\) also lies on \(C\).
    Given that the tangent to \(C\) at \(Q\) is perpendicular to the tangent to \(C\) at \(P\),
  3. show that the \(x\)-coordinate of \(Q\) is \(\frac { 1 } { 3 } ( 2 + \sqrt { 6 } )\).
Edexcel C1 2010 June Q7
6 marks Moderate -0.8
  1. Given that
$$y = 8 x ^ { 3 } - 4 \sqrt { } x + \frac { 3 x ^ { 2 } + 2 } { x } , \quad x > 0$$ find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
(6)
Edexcel C1 2010 June Q11
9 marks Moderate -0.3
  1. The curve \(C\) has equation \(y = \mathrm { f } ( x ) , \quad x > 0\), where
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x - \frac { 5 } { \sqrt { } x } - 2$$ Given that the point \(P ( 4,5 )\) lies on \(C\), find
  1. \(\mathrm { f } ( x )\),
  2. an equation of the tangent to \(C\) at the point \(P\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
Edexcel C1 2011 June Q6
7 marks Easy -1.2
6. Given that \(\frac { 6 x + 3 x ^ { \frac { 5 } { 2 } } } { \sqrt { } x }\) can be written in the form \(6 x ^ { p } + 3 x ^ { q }\),
  1. write down the value of \(p\) and the value of \(q\). Given that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 6 x + 3 x ^ { \frac { 5 } { 2 } } } { \sqrt { } x }\), and that \(y = 90\) when \(x = 4\),
  2. find \(y\) in terms of \(x\), simplifying the coefficient of each term.
Edexcel C1 2011 June Q10
14 marks Moderate -0.3
10. The curve \(C\) has equation $$y = ( x + 1 ) ( x + 3 ) ^ { 2 }$$
  1. Sketch \(C\), showing the coordinates of the points at which \(C\) meets the axes.
  2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } + 14 x + 15\). The point \(A\), with \(x\)-coordinate - 5 , lies on \(C\).
  3. Find the equation of the tangent to \(C\) at \(A\), giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants. Another point \(B\) also lies on \(C\). The tangents to \(C\) at \(A\) and \(B\) are parallel.
  4. Find the \(x\)-coordinate of \(B\).
Edexcel C1 2012 June Q4
6 marks Easy -1.3
4. $$y = 5 x ^ { 3 } - 6 x ^ { \frac { 4 } { 3 } } + 2 x - 3$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) giving each term in its simplest form.
  2. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\)
Edexcel C1 2013 June Q1
4 marks Easy -1.8
Given \(y = x ^ { 3 } + 4 x + 1\), find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(x = 3\)
Edexcel C1 2013 June Q10
10 marks Moderate -0.3
10. A curve has equation \(y = \mathrm { f } ( x )\). The point \(P\) with coordinates \(( 9,0 )\) lies on the curve. Given that $$\mathrm { f } ^ { \prime } ( x ) = \frac { x + 9 } { \sqrt { } x } , \quad x > 0$$
  1. find \(\mathrm { f } ( x )\).
  2. Find the \(x\)-coordinates of the two points on \(y = \mathrm { f } ( x )\) where the gradient of the curve is equal to 10