Differentiation from First Principles

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

Given the function \(f(x) = x - x^2\), defined for all real values of \(x\),

1. Show that \(f'(x) = 1 - 2x\) by differentiating \(f(x)\) from first principles. [4]
2. Find the maximum value of \(f(x)\). [1]
3. Explain why \(f^{-1}(x)\) does not exist. [1]

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

8 A curve has equation \(y = 2 x ^ { 2 }\). The points \(A\) and \(B\) lie on the curve and have \(x\)-coordinates 5 and \(5 + h\) respectively, where \(h > 0\).

1. Show that the gradient of the line \(A B\) is \(20 + 2 h\).
2. Explain how the answer to part (i) relates to the gradient of the curve at \(A\).
3. The normal to the curve at \(A\) meets the \(y\)-axis at the point \(C\). Find the \(y\)-coordinate of \(C\).

7 Differentiate \(\cos x\) with respect to \(x\), from first principles.

3 The points \(\mathrm { P } ( 2,3.6 )\) and \(\mathrm { Q } ( 2.2,2.4 )\) lie on the curve \(y = \mathrm { f } ( x )\). Use P and Q to estimate the gradient of the curve at the point where \(x = 2\).

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

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 \(\mathrm { f } ^ { \prime } ( 2 )\).

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

Differentiate \(f(x) = x^4\) from first principles. [5]

1 The following points $$A ( 0,1 ) , \quad B ( 1,6 ) , \quad C ( 1.5,7.75 ) , \quad D ( 1.9,8.79 ) \quad \text { and } \quad E ( 2,9 )$$ lie on the curve \(y = \mathrm { f } ( x )\). The table below shows the gradients of the chords \(A E\) and \(B E\).

1. Complete the table to show the gradients of \(C E\) and \(D E\).
2. State what the values in the table indicate about the value of \(\mathrm { f } ^ { \prime } ( 2 )\).

The equation of a curve is \(y = \text{f}(x)\), where f\((x) = (2x - 1)\sqrt{3x - 2} - 2\). The following points lie on the curve. Non-exact values have been given correct to 5 decimal places. \(A(2, 4)\), \(B(2.0001, k)\), \(C(2.001, 4.00625)\), \(D(2.01, 4.06261)\), \(E(2.1, 4.63566)\), \(F(3, 11.22876)\)

1. Find the value of \(k\). Give your answer correct to 5 decimal places. [1]

The table shows the gradients of the chords \(AB\), \(AC\), \(AD\) and \(AF\).

1. Find the gradient of the chord \(AE\). Give your answer correct to 4 decimal places. [1]
2. Deduce the value of f\('(2)\) using the values in the table. [1]

5 A curve has equation \(y = x ^ { 3 } - 12 x\).
The point \(A\) on the curve has coordinates ( \(2 , - 16\) ).
The point \(B\) on the curve has \(x\)-coordinate \(2 + h\).

1. Show that the gradient of the line \(A B\) is \(6 h + h ^ { 2 }\).
2. Explain how the result of part (a) can be used to show that \(A\) is a stationary point on the curve.
   \includegraphics[max width=\textwidth, alt={}]{763d89e4-861a-4754-a93c-d0902987673f-06_1894_1709_813_153}

4 The quadratic polynomial \(2 x ^ { 2 } - 3\) is denoted by \(\mathrm { f } ( x )\).
Use differentiation from first principles to determine the value of \(\mathrm { f } ^ { \prime } ( 2 )\).

Jodie is attempting to use differentiation from first principles to prove that the gradient of \(y = \sin x\) is zero when \(x = \frac{\pi}{2}\) Jodie's teacher tells her that she has made mistakes starting in Step 4 of her working. Her working is shown below. \includegraphics{figure_11} Complete Steps 4 and 5 of Jodie's working below, to correct her proof. [4 marks] Step 4 \quad For gradient of curve at A, Step 5 \quad Hence the gradient of the curve at A is given by

5

1. Given that \(\mathrm { f } ( x ) = x ^ { 2 } - 4 x\), use differentiation from first principles to show that \(\mathrm { f } ^ { \prime } ( x ) = 2 x - 4\).
2. Find the equation of the curve through \(( 2,7 )\) for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 x - 4\).

8 Differentiate from first principles $$y = \frac { 1 } { x }$$

2. Differentiate \(f ( x ) = a x ^ { 2 } + b x\) from first principles
(Total for Question 2 is 4 marks)

The function f is defined by \(f(x) = \sqrt{x}, x > 0\).

1. Use differentiation from first principles to find an expression for \(f'(x)\). [5]

The lines \(l_1\) and \(l_2\) are the tangents to the curve \(y = f(x)\) at the points \(A\) and \(B\) where \(x = a\) and \(x = b\) respectively, \(a \neq b\).

1. 1. Show that the tangents intersect at the point \(\left(\sqrt{ab}, \frac{1}{2}(\sqrt{a} + \sqrt{b})\right)\). [5]
   2. Given that \(l_1\) and \(l_2\) intersect at a point with integer coordinates, write down a possible pair of values for \(a\) and \(b\). [2]

1. Calculate the gradient of the chord joining the points on the curve \(y = x^2 - 7\) for which \(x = 3\) and \(x = 3.1\). [2]
2. Given that \(f(x) = x^2 - 7\), find and simplify \(\frac{f(3 + h) - f(3)}{h}\). [3]
3. Use your result in part (ii) to find the gradient of \(y = x^2 - 7\) at the point where \(x = 3\), showing your reasoning. [2]
4. Find the equation of the tangent to the curve \(y = x^2 - 7\) at the point where \(x = 3\). [2]
5. This tangent crosses the \(x\)-axis at the point P. The curve crosses the positive \(x\)-axis at the point Q. Find the distance PQ, giving your answer correct to 3 decimal places. [3]

2 State the value of $$\lim _ { h \rightarrow 0 } \frac { \sin ( \pi + h ) - \sin \pi } { h }$$ Circle your answer. \(\cos h\) -1
0
1