1.07q Product and quotient rules: differentiation

4. (i) $$f ( x ) = \frac { ( 2 x + 5 ) ^ { 2 } } { x - 3 } \quad x \neq 3$$

1. Find \(\mathrm { f } ^ { \prime } ( x )\) in the form \(\frac { P ( x ) } { Q ( x ) }\) where \(P ( x )\) and \(Q ( x )\) are fully factorised quadratic expressions.
2. Hence find the range of values of \(x\) for which \(\mathrm { f } ( x )\) is increasing.
   (ii) $$g ( x ) = x \sqrt { \sin 4 x } \quad 0 \leqslant x < \frac { \pi } { 4 }$$ The curve with equation \(y = g ( x )\) has a maximum at the point \(M\). Show that the \(x\) coordinate of \(M\) satisfies the equation $$\tan 4 x + k x = 0$$ where \(k\) is a constant to be found.
   

6. $$\mathrm { f } ( x ) = x \cos \left( \frac { x } { 3 } \right) \quad x > 0$$

1. Find \(\mathrm { f } ^ { \prime } ( x )\)
2. Show that the equation \(\mathrm { f } ^ { \prime } ( x ) = 0\) can be written as $$x = k \arctan \left( \frac { k } { x } \right)$$ where \(k\) is an integer to be found.
3. Starting with \(x _ { 1 } = 2.5\) use the iteration formula $$x _ { n + 1 } = k \arctan \left( \frac { k } { x _ { n } } \right)$$ with the value of \(k\) found in part (b), to calculate the values of \(x _ { 2 }\) and \(x _ { 6 }\) giving your answers to 3 decimal places.
4. Using a suitable interval and a suitable function that should be stated, show that a root of \(\mathrm { f } ^ { \prime } ( x ) = 0\) is 2.581 correct to 3 decimal places.
   In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

1. Find, using calculus, the \(x\) coordinate of the stationary point on the curve with equation

$$y = ( 2 x + 5 ) e ^ { 3 x }$$

6. \begin{figure}[h] \end{figure} \section*{In this question you must show all stages of your working.} Solutions relying entirely on calculator technology are not acceptable. The function f is defined by $$f ( x ) = 5 \left( x ^ { 2 } - 2 \right) ( 4 x + 9 ) ^ { \frac { 1 } { 2 } } \quad x \geqslant - \frac { 9 } { 4 }$$

1. Show that $$f ^ { \prime } ( x ) = \frac { k \left( 5 x ^ { 2 } + 9 x - 2 \right) } { ( 4 x + 9 ) ^ { \frac { 1 } { 2 } } }$$ where \(k\) is an integer to be found.
2. Hence, find the values of \(x\) for which \(\mathrm { f } ^ { \prime } ( x ) = 0\) Figure 3 shows a sketch of the curve \(C\) with equation \(y = \mathrm { f } ( x )\). The curve has a local maximum at the point \(P\)
3. Find the exact coordinates of \(P\) The function g is defined by $$g ( x ) = 2 f ( x ) + 4 \quad - \frac { 9 } { 4 } \leqslant x \leqslant 0$$
4. Find the range of g

8. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of the curve \(C\) with equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = ( 2 x + 1 ) ^ { 3 } e ^ { - 4 x }$$

1. Show that $$\mathrm { f } ^ { \prime } ( x ) = A ( 2 x + 1 ) ^ { 2 } ( 1 - 4 x ) \mathrm { e } ^ { - 4 x }$$ where \(A\) is a constant to be found.
2. Hence find the exact coordinates of the two stationary points on \(C\). The function g is defined by $$g ( x ) = 8 f ( x - 2 )$$
3. Find the coordinates of the maximum stationary point on the curve with equation \(y = g ( x )\).

9. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = x \left( x ^ { 2 } - 4 \right) e ^ { - \frac { 1 } { 2 } x }$$

1. Find \(f ^ { \prime } ( x )\). The line \(l\) is the normal to the curve at \(O\) and meets the curve again at the point \(P\). The point \(P\) lies in the 3rd quadrant, as shown in Figure 3.
2. Show that the \(x\) coordinate of \(P\) is a solution of the equation $$x = - \frac { 1 } { 2 } \sqrt { 16 + \mathrm { e } ^ { \frac { 1 } { 2 } x } }$$
3. Using the iterative formula $$x _ { n + 1 } = - \frac { 1 } { 2 } \sqrt { 16 + \mathrm { e } ^ { \frac { 1 } { 2 } x _ { n } } } \quad \text { with } x _ { 1 } = - 2$$ find, to 4 decimal places,
   1. the value of \(x _ { 2 }\)
   2. the \(x\) coordinate of \(P\).

3. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = ( x - 2 ) ^ { 2 } \mathrm { e } ^ { 3 x } \quad x \in \mathbb { R }$$ The curve has a maximum turning point at \(A\) and a minimum turning point at \(( 2,0 )\)

1. Use calculus to find the exact coordinates of \(A\). Given that the equation \(\mathrm { f } ( x ) = k\), where \(k\) is a constant, has at least two distinct roots,
2. state the range of possible values for \(k\).

6. $$y = \frac { 2 + 3 \sin x } { \cos x + \sin x }$$ Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { a \tan x + b \sec x + c } { \sec x + 2 \sin x }$$ where \(a , b\) and \(c\) are integers to be found.

1. The curve \(C\) has equation

$$y = \frac { \ln \left( x ^ { 2 } + k \right) } { x ^ { 2 } + k } \quad x \in \mathbb { R }$$ where \(k\) is a positive constant.

1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { A x \left( B - \ln \left( x ^ { 2 } + k \right) \right) } { \left( x ^ { 2 } + k \right) ^ { 2 } }$$ where \(A\) and \(B\) are constants to be found. Given that \(C\) has exactly three turning points,
2. find the \(x\) coordinate of each of these points. Give your answer in terms of \(k\) where appropriate.
3. find the upper limit to the value for \(k\).

7. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve \(C\) with equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = e ^ { - x ^ { 2 } } \left( 2 x ^ { 2 } - 3 \right) ^ { 2 }$$

1. Find the range of f
2. Show that $$\mathrm { f } ^ { \prime } ( x ) = 2 x \left( 2 x ^ { 2 } - 3 \right) \mathrm { e } ^ { - x ^ { 2 } } \left( A - B x ^ { 2 } \right)$$ where \(A\) and \(B\) are constants to be found. Given that the line \(y = k\), where \(k\) is a constant, \(k > 0\), intersects the curve at exactly two distinct points,
3. find the exact range of values of \(k\)

5. Given that $$y = \frac { 5 x ^ { 2 } - 10 x + 9 } { ( x - 1 ) ^ { 2 } } \quad x \neq 1$$ show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { k } { ( x - 1 ) ^ { 3 } }\), where \(k\) is a constant to be found.
(6)

1. $$\mathrm { f } ( x ) = \frac { 2 x } { x ^ { 2 } + 3 } , \quad x \in \mathbb { R }$$ Find the set of values of \(x\) for which \(\mathrm { f } ^ { \prime } ( x ) > 0\) You must show your working.
(Solutions based entirely on graphical or numerical methods are not acceptable.)

7. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = 2 x ( 1 + x ) \ln x , \quad x > 0$$ The curve has a minimum turning point at \(A\).

1. Find f'(x)
2. Hence show that the \(x\) coordinate of \(A\) is the solution of the equation $$x = \mathrm { e } ^ { - \frac { 1 + x } { 1 + 2 x } }$$
3. Use the iteration formula $$x _ { n + 1 } = \mathrm { e } ^ { - \frac { 1 + x _ { n } } { 1 + 2 x _ { n } } } , \quad x _ { 0 } = 0.46$$ to find the values of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\) to 4 decimal places.
4. Use your answer to part (c) to estimate the coordinates of \(A\) to 2 decimal places.

1. The curve \(C\) has equation

$$y = \frac { 3 x - 2 } { ( x - 2 ) ^ { 2 } } , \quad x \neq 2$$ The point \(P\) on \(C\) has \(x\) coordinate 3
Find an equation of the normal to \(C\) at the point \(P\) in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

7. \begin{figure}[h] \end{figure} Figure 3 shows part of the curve \(C\) with equation $$y = \frac { 3 \ln \left( x ^ { 2 } + 1 \right) } { \left( x ^ { 2 } + 1 \right) } , \quad x \in \mathbb { R }$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
2. Using your answer to (a), find the exact coordinates of the stationary point on the curve \(C\) for which \(x > 0\). Write each coordinate in its simplest form.
   (5) The finite region \(R\), shown shaded in Figure 3, is bounded by the curve \(C\), the \(x\)-axis and the line \(x = 3\)
3. Complete the table below with the value of \(y\) corresponding to \(x = 1\)

   \(x\)	0	1	2	3
   \(y\)	0		\(\frac { 3 } { 5 } \ln 5\)	\(\frac { 3 } { 10 } \ln 10\)

4. Use the trapezium rule with all the \(y\) values in the completed table to find an approximate value for the area of \(R\), giving your answer to 4 significant figures.

1. (i) Differentiate \(y = 5 x ^ { 2 } \ln 3 x , \quad x > 0\) (ii) Given that

$$y = \frac { x } { \sin x + \cos x } , \quad - \frac { \pi } { 4 } < x < \frac { 3 \pi } { 4 }$$ show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { ( 1 + x ) \sin x + ( 1 - x ) \cos x } { 1 + \sin 2 x } , \quad - \frac { \pi } { 4 } < x < \frac { 3 \pi } { 4 }$$ \includegraphics[max width=\textwidth, alt={}, center]{e30f0c28-1695-40a1-8e9a-6ea7e29042bf-11_99_104_2631_1781}

14. Given that $$y = \frac { \left( x ^ { 2 } - 4 \right) ^ { \frac { 1 } { 2 } } } { x ^ { 3 } } \quad x > 2$$

1. show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { A x ^ { 2 } + 12 } { x ^ { 4 } \left( x ^ { 2 } - 4 \right) ^ { \frac { 1 } { 2 } } } \quad x > 2$$ where \(A\) is a constant to be found. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{a377da06-a968-438c-bec2-ae55283dae47-48_593_1134_865_395} \captionsetup{labelformat=empty} \caption{Figure 4}
   \end{figure} Figure 4 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = \frac { 24 \left( x ^ { 2 } - 4 \right) ^ { \frac { 1 } { 2 } } } { x ^ { 3 } } \quad x > 2$$
2. Use your answer to part (a) to find the range of f.
3. State a reason why f-1 does not exist.

8. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the curve \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = \frac { 6 x + 2 } { 3 x ^ { 2 } + 5 } , \quad x \in \mathbb { R }$$

1. Find \(\mathrm { f } ^ { \prime } ( x )\), writing your answer as a single fraction in its simplest form. The curve has two turning points, a maximum at point \(A\) and a minimum at point \(B\), as shown in Figure 2.
2. Using part (a), find the coordinates of point \(A\) and the coordinates of point \(B\).
3. State the coordinates of the maximum turning point of the function with equation $$y = \mathrm { f } ( 2 x ) + 4 \quad x \in \mathbb { R }$$
4. Find the range of the function $$\operatorname { g } ( x ) = \frac { 6 x + 2 } { 3 x ^ { 2 } + 5 } , \quad x \leqslant 0$$

4. (a) Differentiate with respect to \(x\)

1. \(x ^ { 2 } \mathrm { e } ^ { 3 x + 2 }\),
2. \(\frac { \cos \left( 2 x ^ { 3 } \right) } { 3 x }\).
   (b) Given that \(x = 4 \sin ( 2 y + 6 )\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\).

4. (i) The curve \(C\) has equation $$y = \frac { x } { 9 + x ^ { 2 } }$$ Use calculus to find the coordinates of the turning points of \(C\).
(ii) Given that $$y = \left( 1 + \mathrm { e } ^ { 2 x } \right) ^ { \frac { 3 } { 2 } }$$ find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at \(x = \frac { 1 } { 2 } \ln 3\).

2. A curve \(C\) has equation $$y = \mathrm { e } ^ { 2 x } \tan x , \quad x \neq ( 2 n + 1 ) \frac { \pi } { 2 }$$

1. Show that the turning points on \(C\) occur where \(\tan x = - 1\).
2. Find an equation of the tangent to \(C\) at the point where \(x = 0\).

1. The functions \(f\) and \(g\) are defined by

$$\begin{aligned} & \mathrm { f } : x \mapsto 1 - 2 x ^ { 3 } , x \in \mathbb { R } \\ & \mathrm {~g} : x \mapsto \frac { 3 } { x } - 4 , x > 0 , x \in \mathbb { R } \end{aligned}$$

1. Find the inverse function \(\mathrm { f } ^ { - 1 }\).
2. Show that the composite function gf is $$\text { gf } : x \mapsto \frac { 8 x ^ { 3 } - 1 } { 1 - 2 x ^ { 3 } }$$
3. Solve \(\operatorname { gf } ( x ) = 0\).
4. Use calculus to find the coordinates of the stationary point on the graph of \(y = \operatorname { gf } ( x )\).

1. (a) Find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at the point where \(x = 2\) on the curve with equation

$$y = x ^ { 2 } \sqrt { } ( 5 x - 1 )$$ (b) Differentiate \(\frac { \sin 2 x } { x ^ { 2 } }\) with respect to \(x\).

Differentiate with respect to \(x\), giving your answer in its simplest form,

1. \(x ^ { 2 } \ln ( 3 x )\)
2. \(\frac { \sin 4 x } { x ^ { 3 } }\)

5. (i) Differentiate with respect to \(x\)

1. \(y = x ^ { 3 } \ln 2 x\)
2. \(y = ( x + \sin 2 x ) ^ { 3 }\) Given that \(x = \cot y\),
   (ii) show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { - 1 } { 1 + x ^ { 2 } }\)