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

726 questions

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Edexcel C1 Q7
10 marks Moderate -0.8
The curve \(C\) has equation \(y = 4x^2 + \frac{5-x}{x}\), \(x \neq 0\). The point \(P\) on \(C\) has \(x\)-coordinate \(1\).
  1. Show that the value of \(\frac{dy}{dx}\) at \(P\) is \(3\). [5]
  2. Find an equation of the tangent to \(C\) at \(P\). [3]
This tangent meets the \(x\)-axis at the point \((k, 0)\).
  1. Find the value of \(k\). [2]
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 Q10
11 marks Moderate -0.8
The curve \(C\) has equation \(y = \frac{1}{3}x^3 - 4x^2 + 8x + 3\). The point \(P\) has coordinates \((3, 0)\).
  1. Show that \(P\) lies on \(C\). [1]
  2. Find the equation of the tangent to \(C\) at \(P\), giving your answer in the form \(y = mx + c\), where \(m\) and \(c\) are constants. [5]
Another point \(Q\) also lies on \(C\). The tangent to \(C\) at \(Q\) is parallel to the tangent to \(C\) at \(P\).
  1. Find the coordinates of \(Q\). [5]
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 Q9
12 marks Moderate -0.8
\includegraphics{figure_2} Figure 2 shows part of the curve \(C\) with equation $$y = (x - 1)(x^2 - 4).$$ The curve cuts the \(x\)-axis at the points \(P\), \((1, 0)\) and \(Q\), as shown in Figure 2.
  1. Write down the \(x\)-coordinate of \(P\) and the \(x\)-coordinate of \(Q\). [2]
  2. Show that \(\frac{dy}{dx} = 3x^2 - 2x - 4\). [3]
  3. Show that \(y = x + 7\) is an equation of the tangent to \(C\) at the point \((-1, 6)\). [2]
The tangent to \(C\) at the point \(R\) is parallel to the tangent at the point \((-1, 6)\).
  1. Find the exact coordinates of \(R\). [5]
Edexcel C1 Q5
7 marks Easy -1.2
Differentiate with respect to \(x\)
  1. \(x^4 + 6\sqrt{x}\), [3]
  2. \(\frac{(x + 4)^3}{x}\). [4]
Edexcel C1 Q10
10 marks Moderate -0.8
The curve \(C\) with equation \(y = f(x)\), \(x \neq 0\), passes through the point \((3, 7\frac{1}{2})\). Given that \(f'(x) = 2x + \frac{3}{x^2}\),
  1. find \(f(x)\). [5]
  2. Verify that \(f(-2) = 5\). [1]
  3. Find an equation for the tangent to \(C\) at the point \((-2, 5)\), giving your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [4]
Edexcel C1 Q1
4 marks Easy -1.2
Given that $$y = 4x^3 - 1 + 2x^{-1}, \quad x > 0,$$ find \(\frac{dy}{dx}\). [4]
Edexcel C1 Q7
9 marks Moderate -0.3
The curve \(C\) has equation \(y = f(x)\), \(x \neq 0\), and the point \(P(2, 1)\) lies on \(C\). Given that $$f'(x) = 3x^2 - 6 - \frac{8}{x^3},$$
  1. find \(f(x)\). [5]
  2. Find an equation for the tangent to \(C\) at the point \(P\), giving your answer in the form \(y = mx + c\), where \(m\) and \(c\) are integers. [4]
Edexcel C1 Q8
11 marks Moderate -0.8
The curve \(C\) has equation \(y = 4x + 3x^{-1} - 2x^2\), \(x > 0\).
  1. Find an expression for \(\frac{dy}{dx}\). [3]
  2. Show that the point \(P(4, 8)\) lies on \(C\). [1]
  3. Show that an equation of the normal to \(C\) at the point \(P\) is $$3y - x + 20.$$ [4]
The normal to \(C\) at \(P\) cuts the \(x\)-axis at the point \(Q\).
  1. Find the length \(PQ\), giving your answer in a simplified surd form. [3]
Edexcel C1 Q11
11 marks Moderate -0.8
A curve \(C\) has equation \(y = x^3 - 5x^2 + 5x + 2\).
  1. Find \(\frac{dy}{dx}\) in terms of \(x\). [2]
The points \(P\) and \(Q\) lie on \(C\). The gradient of \(C\) at both \(P\) and \(Q\) is 2. The \(x\)-coordinate of \(P\) is 3.
  1. Find the \(x\)-coordinate of \(Q\). [2]
  2. Find an equation for the tangent to \(C\) at \(P\), giving your answer in the form \(y = mx + c\), where \(m\) and \(c\) are constants. [3]
This tangent intersects the coordinate axes at the points \(R\) and \(S\).
  1. Find the length of \(RS\), giving your answer as a surd. [4]
Edexcel C1 Q21
5 marks Easy -1.3
\(y = 7 + 10x^{\frac{3}{2}}\).
  1. Find \(\frac{dy}{dx}\). [2]
  2. Find \(\int y \, dx\). [3]
Edexcel C1 Q28
7 marks Moderate -0.8
For the curve \(C\) with equation \(y = x^4 - 8x^2 + 3\),
  1. find \(\frac{dy}{dx}\). [2]
The point \(A\), on the curve \(C\), has \(x\)-coordinate 1.
  1. Find an equation for the normal to \(C\) at \(A\), giving your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [5]
Edexcel C1 Q36
9 marks Moderate -0.8
The curve \(C\) with equation \(y = f(x)\) is such that $$\frac{dy}{dx} = 3\sqrt{x} + \frac{12}{\sqrt{x}}, \quad x > 0.$$
  1. Show that, when \(x = 8\), the exact value of \(\frac{dy}{dx}\) is \(9\sqrt{2}\). [3]
The curve \(C\) passes through the point \((4, 30)\).
  1. Using integration, find \(f(x)\). [6]
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 C1 Specimen Q9
11 marks Easy -1.2
The curve \(C\) has equation \(y = \text{f}(x)\) and the point \(P(3, 5)\) lies on \(C\). Given that $$\text{f}(x) = 3x^2 - 8x + 6,$$
  1. find \(\text{f}'(x)\). [4]
  2. Verify that the point \((2, 0)\) lies on \(C\). [2]
The point \(Q\) also lies on \(C\), and the tangent to \(C\) at \(Q\) is parallel to the tangent to \(C\) at \(P\).
  1. Find the \(x\)-coordinate of \(Q\). [5]
Edexcel C1 Specimen Q10
13 marks Moderate -0.8
The curve \(C\) has equation \(y = x^3 - 5x + \frac{2}{x}\), \(x \neq 0\). The points \(A\) and \(B\) both lie on \(C\) and have coordinates \((1, -2)\) and \((-1, 2)\) respectively.
  1. Show that the gradient of \(C\) at \(A\) is equal to the gradient of \(C\) at \(B\). [5]
  2. Show that an equation for the normal to \(C\) at \(A\) is \(4y = x - 9\). [4]
The normal to \(C\) at \(A\) meets the \(y\)-axis at the point \(P\). The normal to \(C\) at \(B\) meets the \(y\)-axis at the point \(Q\).
  1. Find the length of \(PQ\). [4]
Edexcel P2 2022 June Q8
8 marks Standard +0.3
In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. A curve has equation $$y = 256x^4 - 304x - 35 + \frac{27}{x^2} \quad x \neq 0$$
  1. Find \(\frac{dy}{dx}\) [3]
  2. Hence find the coordinates of the stationary points of the curve. [5]
Edexcel C2 Q7
10 marks Moderate -0.8
The curve \(C\) has equation \(y = 2x^3 - 5x^2 - 4x + 2\).
  1. Find \(\frac{dy}{dx}\). [2]
  2. Using the result from part (a), find the coordinates of the turning points of \(C\). [4]
  3. Find \(\frac{d^2y}{dx^2}\). [2]
  4. Hence, or otherwise, determine the nature of the turning points of \(C\). [2]
Edexcel C2 Q10
11 marks Standard +0.3
\includegraphics{figure_4} Figure 4 shows a solid brick in the shape of a cuboid measuring 2x cm by x cm by y cm. The total surface area of the brick is 600 cm².
  1. Show that the volume, \(V\) cm³, of the brick is given by \(V = 200x - \frac{4x^3}{3}\). [4]
Given that \(x\) can vary,
  1. use calculus to find the maximum value of \(V\), giving your answer to the nearest cm³. [5]
  2. Justify that the value of \(V\) you have found is a maximum. [2]
Edexcel C2 Q25
11 marks Standard +0.3
Given that \(f'(x) = (2x^3 - 3x^{-2})^2 + 5\), \(x > 0\),
  1. Find, to 3 significant figures, the value of \(x\) for which \(f'(x) = 5\). [3]
  2. Show that \(f'(x)\) may be written in the form \(Ax^6 + \frac{B}{x^4} + C\), where \(A\), \(B\) and \(C\) are constants to be found. [3]
  3. Hence evaluate \(\int_1^2 f'(x) \, dx\). [5]
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 C2 Q38
10 marks Moderate -0.3
On a journey, the average speed of a car is \(v\) m s\(^{-1}\). For \(v \geq 5\), the cost per kilometre, \(C\) pence, of the journey is modelled by $$C = \frac{160}{v} + \frac{v^2}{100}.$$ Using this model,
  1. show, by calculus, that there is a value of \(v\) for which \(C\) has a stationary value, and find this value of \(v\). [5]
  2. Justify that this value of \(v\) gives a minimum value of \(C\). [2]
  3. Find the minimum value of \(C\) and hence find the minimum cost of a 250 km car journey. [3]
Edexcel C3 Q1
10 marks Moderate -0.3
The function f, defined for \(x \in \mathbb{R}, x > 0\), is such that $$f'(x) = x^2 - 2 + \frac{1}{x^2}.$$
  1. Find the value of \(f''(x)\) at \(x = 4\). [3]
  2. Given that \(f(3) = 0\), find \(f(x)\). [4]
  3. Prove that \(f\) is an increasing function. [3]
Edexcel C3 Q2
5 marks Moderate -0.8
The curve \(C\) has equation \(y = 2e^x + 3x^2 + 2\). The point \(A\) with coordinates \((0, 4)\) lies on \(C\). Find the equation of the tangent to \(C\) at \(A\). [5]