1.07q Product and quotient rules: differentiation

7. $$\mathrm { h } ( x ) = \frac { 2 } { x + 2 } + \frac { 4 } { x ^ { 2 } + 5 } - \frac { 18 } { \left( x ^ { 2 } + 5 \right) ( x + 2 ) } , \quad x \geqslant 0$$

1. Show that \(\mathrm { h } ( x ) = \frac { 2 x } { x ^ { 2 } + 5 }\)
2. Hence, or otherwise, find \(\mathrm { h } ^ { \prime } ( x )\) in its simplest form. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c78b0245-5c5a-407f-ad8a-602949a76e05-10_729_1235_644_351} \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{figure} Figure 2 shows a graph of the curve with equation \(y = \mathrm { h } ( x )\).
3. Calculate the range of \(\mathrm { h } ( x )\).

2. $$f ( x ) = \frac { 15 } { 3 x + 4 } - \frac { 2 x } { x - 1 } + \frac { 14 } { ( 3 x + 4 ) ( x - 1 ) } , \quad x > 1$$

1. Express \(\mathrm { f } ( x )\) as a single fraction in its simplest form.
2. Hence, or otherwise, find \(\mathrm { f } ^ { \prime } ( x )\), giving your answer as a single fraction in its simplest form.

1. (a) By writing \(\operatorname { cosec } x\) as \(\frac { 1 } { \sin x }\), show that

$$\frac { \mathrm { d } ( \operatorname { cosec } x ) } { \mathrm { d } x } = - \operatorname { cosec } x \cot x$$ Given that \(y = \mathrm { e } ^ { 3 x } \operatorname { cosec } 2 x , 0 < x < \frac { \pi } { 2 }\),
(b) find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\). The curve with equation \(y = \mathrm { e } ^ { 3 x } \operatorname { cosec } 2 x , 0 < x < \frac { \pi } { 2 }\), has a single turning point.
(c) Show that the \(x\)-coordinate of this turning point is at \(x = \frac { 1 } { 2 } \arctan k\) where the value
of the constant \(k\) should be found. of the constant \(k\) should be found.

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

1. \(3 \sin ^ { 2 } x + \sec 2 x\),
2. \(\{ x + \ln ( 2 x ) \} ^ { 3 }\). Given that \(y = \frac { 5 x ^ { 2 } - 10 x + 9 } { ( x - 1 ) ^ { 2 } } , \quad x \neq 1\),
   (b) show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 8 } { ( x - 1 ) ^ { 3 } }\).

3. A curve \(C\) has equation $$y = x ^ { 2 } \mathrm { e } ^ { x }$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), using the product rule for differentiation.
2. Hence find the coordinates of the turning points of \(C\).
3. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
4. Determine the nature of each turning point of the curve \(C\).

6. (a) Differentiate with respect to \(x\),

1. \(\mathrm { e } ^ { 3 x } ( \sin x + 2 \cos x )\),
2. \(x ^ { 3 } \ln ( 5 x + 2 )\). Given that \(y = \frac { 3 x ^ { 2 } + 6 x - 7 } { ( x + 1 ) ^ { 2 } } , \quad x \neq - 1\),
   (b) show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 20 } { ( x + 1 ) ^ { 3 } }\).
   (c) Hence find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) and the real values of \(x\) for which \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = - \frac { 15 } { 4 }\).

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

1. \(x ^ { 2 } \cos 3 x\)
2. \(\frac { \ln \left( x ^ { 2 } + 1 \right) } { x ^ { 2 } + 1 }\) (ii) A curve \(C\) has the equation $$y = \sqrt { } ( 4 x + 1 ) , \quad x > - \frac { 1 } { 4 } , \quad y > 0$$ The point \(P\) on the curve has \(x\)-coordinate 2 . Find an equation of the tangent to \(C\) at \(P\) in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

5. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve \(C\) with the equation \(y = \left( 2 x ^ { 2 } - 5 x + 2 \right) \mathrm { e } ^ { - x }\).

1. Find the coordinates of the point where \(C\) crosses the \(y\)-axis.
2. Show that \(C\) crosses the \(x\)-axis at \(x = 2\) and find the \(x\)-coordinate of the other point where \(C\) crosses the \(x\)-axis.
3. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
4. Hence find the exact coordinates of the turning points of \(C\).

Differentiate with respect to \(x\)

1. \(\quad \ln \left( x ^ { 2 } + 3 x + 5 \right)\)
2. \(\frac { \cos x } { x ^ { 2 } }\)

1. (a) Express \(2 \cos 3 x - 3 \sin 3 x\) in the form \(R \cos ( 3 x + \alpha )\), where \(R\) and \(\alpha\) are constants, \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\). Give your answers to 3 significant figures.

$$\mathrm { f } ( x ) = \mathrm { e } ^ { 2 x } \cos 3 x$$ (b) Show that \(\mathrm { f } ^ { \prime } ( x )\) can be written in the form $$\mathrm { f } ^ { \prime } ( x ) = R \mathrm { e } ^ { 2 x } \cos ( 3 x + \alpha )$$ where \(R\) and \(\alpha\) are the constants found in part (a).
(c) Hence, or otherwise, find the smallest positive value of \(x\) for which the curve with equation \(y = \mathrm { f } ( x )\) has a turning point.

5. (a) Differentiate $$\frac { \cos 2 x } { \sqrt { x } }$$ with respect to \(x\).
(b) Show that \(\frac { \mathrm { d } } { \mathrm { d } x } \left( \sec ^ { 2 } 3 x \right)\) can be written in the form $$\mu \left( \tan 3 x + \tan ^ { 3 } 3 x \right)$$ where \(\mu\) is a constant.
(c) Given \(x = 2 \sin \left( \frac { y } { 3 } \right)\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\), simplifying your answer.

7. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = \left( x ^ { 2 } + 3 x + 1 \right) \mathrm { e } ^ { x ^ { 2 } }$$ The curve cuts the \(x\)-axis at points \(A\) and \(B\) as shown in Figure 2 .

1. Calculate the \(x\) coordinate of \(A\) and the \(x\) coordinate of \(B\), giving your answers to 3 decimal places.
2. Find \(\mathrm { f } ^ { \prime } ( x )\). The curve has a minimum turning point at the point \(P\) as shown in Figure 2.
3. Show that the \(x\) coordinate of \(P\) is the solution of $$x = - \frac { 3 \left( 2 x ^ { 2 } + 1 \right) } { 2 \left( x ^ { 2 } + 2 \right) }$$
4. Use the iteration formula $$x _ { n + 1 } = - \frac { 3 \left( 2 x _ { n } ^ { 2 } + 1 \right) } { 2 \left( x _ { n } ^ { 2 } + 2 \right) } , \quad \text { with } x _ { 0 } = - 2.4$$ to calculate the values of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\), giving your answers to 3 decimal places. The \(x\) coordinate of \(P\) is \(\alpha\).
5. By choosing a suitable interval, prove that \(\alpha = - 2.43\) to 2 decimal places.

1. (i) (a) Show that \(\frac { \mathrm { d } } { \mathrm { d } x } \left( x ^ { \frac { 1 } { 2 } } \ln x \right) = \frac { \ln x } { 2 \sqrt { } x } + \frac { 1 } { \sqrt { } x }\)

The curve with equation \(y = x ^ { \frac { 1 } { 2 } } \ln x , x > 0\) has one turning point at the point \(P\).
(b) Find the exact coordinates of \(P\). Give your answer in its simplest form.
(ii) A curve \(C\) has equation \(y = \frac { x - k } { x + k }\), where \(k\) is a positive constant. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), and show that \(C\) has no turning points.

4. $$\mathrm { f } ( x ) = 25 x ^ { 2 } \mathrm { e } ^ { 2 x } - 16 , \quad x \in \mathbb { R }$$

1. Using calculus, find the exact coordinates of the turning points on the curve with equation \(y = \mathrm { f } ( x )\).
2. Show that the equation \(\mathrm { f } ( x ) = 0\) can be written as \(x = \pm \frac { 4 } { 5 } \mathrm { e } ^ { - x }\) The equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\), where \(\alpha = 0.5\) to 1 decimal place.
3. Starting with \(x _ { 0 } = 0.5\), use the iteration formula $$x _ { n + 1 } = \frac { 4 } { 5 } \mathrm { e } ^ { - x _ { n } }$$ to calculate the values of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\), giving your answers to 3 decimal places.
4. Give an accurate estimate for \(\alpha\) to 2 decimal places, and justify your answer.

1. (i) Given that

$$x = \sec ^ { 2 } 2 y , \quad 0 < y < \frac { \pi } { 4 }$$ show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 4 x \sqrt { ( x - 1 ) } }$$ (ii) Given that $$y = \left( x ^ { 2 } + x ^ { 3 } \right) \ln 2 x$$ find the exact value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at \(x = \frac { \mathrm { e } } { 2 }\), giving your answer in its simplest form.
(iii) Given that $$f ( x ) = \frac { 3 \cos x } { ( x + 1 ) ^ { \frac { 1 } { 3 } } } , \quad x \neq - 1$$ show that $$\mathrm { f } ^ { \prime } ( x ) = \frac { \mathrm { g } ( x ) } { ( x + 1 ) ^ { \frac { 4 } { 3 } } } , \quad x \neq - 1$$ where \(\mathrm { g } ( x )\) is an expression to be found.

1. The curve \(C\) has equation \(y = \mathrm { f } ( x )\) where

$$f ( x ) = \frac { 4 x + 1 } { x - 2 } , \quad x > 2$$

1. Show that $$f ^ { \prime } ( x ) = \frac { - 9 } { ( x - 2 ) ^ { 2 } }$$ Given that \(P\) is a point on \(C\) such that \(\mathrm { f } ^ { \prime } ( x ) = - 1\),
2. find the coordinates of \(P\).

7. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the curve with equation $$\mathrm { g } ( x ) = x ^ { 2 } ( 1 - x ) \mathrm { e } ^ { - 2 x } , \quad x \geqslant 0$$

1. Show that \(\mathrm { g } ^ { \prime } ( x ) = \mathrm { f } ( x ) \mathrm { e } ^ { - 2 x }\), where \(\mathrm { f } ( x )\) is a cubic function to be found.
2. Hence find the range of g .
3. State a reason why the function \(\mathrm { g } ^ { - 1 } ( x )\) does not exist.

2.

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), writing your answer as a single fraction in its simplest form.
2. Hence find the set of values of \(x\) for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } < 0\) 2. $$y = \frac { 4 x } { x ^ { 2 } + 5 }$$

5. (i) Find, using calculus, the \(x\) coordinate of the turning point of the curve with equation $$y = \mathrm { e } ^ { 3 x } \cos 4 x , \quad \frac { \pi } { 4 } \leqslant x < \frac { \pi } { 2 }$$ Give your answer to 4 decimal places.
(ii) Given \(x = \sin ^ { 2 } 2 y , \quad 0 < y < \frac { \pi } { 4 }\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) as a function of \(y\). Write your answer in the form $$\frac { \mathrm { d } y } { \mathrm {~d} x } = p \operatorname { cosec } ( q y ) , \quad 0 < y < \frac { \pi } { 4 }$$ where \(p\) and \(q\) are constants to be determined. \includegraphics[max width=\textwidth, alt={}, center]{d3ba2776-eedb-48f0-834f-41aa454afba3-09_2258_47_315_37}

4. Differentiate with respect to \(x\)

1. \(x ^ { 3 } \mathrm { e } ^ { 3 x }\),
2. \(\frac { 2 x } { \cos x }\),
3. \(\tan ^ { 2 } x\). Given that \(x = \cos y ^ { 2 }\),
4. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(y\).

5. $$y = \mathrm { e } ^ { \cos ^ { 2 } x }$$

1. Show that $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \mathrm { e } ^ { \cos ^ { 2 } x } \left( \sin ^ { 2 } 2 x - 2 \cos 2 x \right)$$
2. Hence find the Maclaurin series expansion of \(\mathrm { e } ^ { \cos ^ { 2 } x }\) up to and including the term in \(x ^ { 2 }\)

6. A curve \(C\) has equation $$y = x ^ { \sin x } \quad x > 0 \quad y > 0$$

1. Find, by firstly taking natural logarithms, an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
2. Hence show that the \(x\) coordinates of the stationary points of \(C\) are solutions of the equation $$\tan x + x \ln x = 0$$