1.07g Differentiation from first principles: for small positive integer powers of x

7 [Figure 1, printed on the insert, is provided for use in this question.]
The diagram shows the curve $$y = x ^ { 3 } - x + 1$$ The points \(A\) and \(B\) on the curve have \(x\)-coordinates - 1 and \(- 1 + h\) respectively. \includegraphics[max width=\textwidth, alt={}, center]{a0a30197-ca11-40d9-9ccd-30281c5e0fb4-05_978_1184_676_411}

1. 1. Show that the \(y\)-coordinate of the point \(B\) is $$1 + 2 h - 3 h ^ { 2 } + h ^ { 3 }$$
   2. Find the gradient of the chord \(A B\) in the form $$p + q h + r h ^ { 2 }$$ where \(p , q\) and \(r\) are integers.
   3. Explain how your answer to part (a)(ii) can be used to find the gradient of the tangent to the curve at \(A\). State the value of this gradient.
2. The equation \(x ^ { 3 } - x + 1 = 0\) has one real root, \(\alpha\).
   1. Taking \(x _ { 1 } = - 1\) as a first approximation to \(\alpha\), use the Newton-Raphson method to find a second approximation, \(x _ { 2 }\), to \(\alpha\).
   2. On Figure 1, draw a straight line to illustrate the Newton-Raphson method as used in part (b)(i). Show the points \(\left( x _ { 2 } , 0 \right)\) and \(( \alpha , 0 )\) on your diagram.

4 The function f is defined for all real values of \(x\) by $$\mathrm { f } ( x ) = x ^ { 3 } + x$$

1. Express \(\mathrm { f } ( 2 + h ) - \mathrm { f } ( 2 )\) in the form $$p h + q h ^ { 2 } + r h ^ { 3 }$$ where \(p , q\) and \(r\) are integers.
2. Use your answer to part (a) to find the value of \(\mathrm { f } ^ { \prime } ( 2 )\).

8

1. The function f is defined for all real values of \(x\) by $$\mathrm { f } ( x ) = x ^ { 3 } + x ^ { 2 } - 1$$
   1. Express \(\mathrm { f } ( 1 + h ) - \mathrm { f } ( 1 )\) in the form $$p h + q h ^ { 2 } + r h ^ { 3 }$$ where \(p , q\) and \(r\) are integers.
   2. Use your answer to part (a)(i) to find the value of \(f ^ { \prime } ( 1 )\).
2. The diagram shows the graphs of $$y = \frac { 1 } { x ^ { 2 } } \quad \text { and } \quad y = x + 1 \quad \text { for } \quad x > 0$$
   \includegraphics[max width=\textwidth, alt={}]{e44987a7-2cdf-442a-aecb-abd3e889ecd4-5_643_791_1160_596}
   The graphs intersect at the point \(P\).
   1. Show that the \(x\)-coordinate of \(P\) satisfies the equation \(\mathrm { f } ( x ) = 0\), where f is the function defined in part (a).
   2. Taking \(x _ { 1 } = 1\) as a first approximation to the root of the equation \(\mathrm { f } ( x ) = 0\), use the Newton-Raphson method to find a second approximation \(x _ { 2 }\) to the root.
      (3 marks)
3. The region enclosed by the curve \(y = \frac { 1 } { x ^ { 2 } }\), the line \(x = 1\) and the \(x\)-axis is shaded on the diagram. By evaluating an improper integral, find the area of this region.
   (3 marks)

8 It is given that \(y = 3 x - 5 x ^ { 2 }\) Use differentiation from first principles to find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) [0pt] [4 marks]
LIH

12

1. Prove the identity \(\frac { 1 } { ( x + h ) ^ { 2 } } - \frac { 1 } { x ^ { 2 } } \equiv \frac { - 2 h x - h ^ { 2 } } { x ^ { 2 } ( x + h ) ^ { 2 } }\).
2. Given that \(\mathrm { f } ( x ) = x ^ { - 2 }\), use differentiation from first principles to find an expression for \(\mathrm { f } ^ { \prime } ( x )\).

3 Given that \(\mathrm { f } ( x ) = x ^ { 3 }\), use differentiation from first principles to prove that \(\mathrm { f } ^ { \prime } ( x ) = 3 x ^ { 2 }\).

A curve \(C\) has equation \(y = x(x + 3)\).

1. Find the gradient of the line passing through the point \((-5, 10)\) and the point on \(C\) with \(x\)-coordinate \(-5 + h\). Give your answer in its simplest form. [3 marks]
2. Show how the answer to part (a) can be used to find the gradient of the curve \(C\) at the point \((-5, 10)\). State the value of this gradient. [2 marks]

A curve \(C\) has equation \(y = (2 - x)(1 + x) + 3\).

1. A line passes through the point \((2, 3)\) and the point on \(C\) with \(x\)-coordinate \(2 + h\). Find the gradient of the line, giving your answer in its simplest form. [3 marks]
2. Show how your answer to part (a) can be used to find the gradient of the curve \(C\) at the point \((2, 3)\). State the value of this gradient. [2 marks]

Craig is investigating the gradient of chords of the curve with equation \(\mathrm{f}(x) = x - x^2\) Each chord joins the point \((3, -6)\) to the point \((3 + h, \mathrm{f}(3 + h))\) The table shows some of Craig's results.

\(x\)	\(\mathrm{f}(x)\)	\(h\)	\(x + h\)	\(\mathrm{f}(x + h)\)	Gradient
\(3\)	\(-6\)	\(1\)	\(4\)	\(-12\)	\(-6\)
\(3\)	\(-6\)	\(0.1\)	\(3.1\)	\(-6.51\)	\(-5.1\)
\(3\)	\(-6\)	\(0.01\)
\(3\)	\(-6\)	\(0.001\)
\(3\)	\(-6\)	\(0.0001\)

1. Show how the value \(-5.1\) has been calculated. [1 mark]
2. Complete the third row of the table above. [2 marks]
3. State the limit suggested by Craig's investigation for the gradient of these chords as \(h\) tends to \(0\) [1 mark]
4. Using differentiation from first principles, verify that your result in part (c) is correct. [4 marks]

Differentiate from first principles $$y = 4x^2 + x$$ [4 marks]

Points \(P\) and \(Q\) lie on the curve with equation \(y = x^4\) The \(x\)-coordinate of \(P\) is \(x\) The \(x\)-coordinate of \(Q\) is \(x + h\)

1. Expand \((x + h)^4\) [2 marks]
2. Hence, find an expression, in terms of \(x\) and \(h\), for the gradient of the line \(PQ\) [1 mark]
3. Explain how to use the answer from part (b) to obtain the gradient function of \(y = x^4\) [2 marks]

\(f(x) = \sin x\) Using differentiation from first principles find the exact value of \(f'\left(\frac{\pi}{6}\right)\) Fully justify your answer. [6 marks]

\(f(x) = 3x^2\) Obtain \(\lim_{h \to 0} \frac{f(x + h) - f(x)}{h}\) Circle your answer. [1 mark] \(\frac{3h^2}{h}\) \quad \(x^3\) \quad \(\frac{3(x + h)^2 - 3x^2}{h}\) \quad \(6x\)

It is given that $$f'(x) = 5x^3 + x$$ Use differentiation from first principles to prove that $$f''(x) = 15x^2 + 1$$ [5 marks]

Prove, from first principles, that the derivative of \(3x^2\) is \(6x\). [4]

Prove, from the first principles, that the derivative of \(5x^2\) is \(10x\).

Given that \(y = x^3\), find \(\frac{dy}{dx}\) from first principles. [6]