Show derivative equals given algebraic form

6 \includegraphics[max width=\textwidth, alt={}, center]{76df3465-9617-4f2b-a8b7-f474b2817504-10_417_700_310_685} The diagram shows the curve with equation \(y = \frac { \ln ( 2 x + 1 ) } { x + 3 }\). The curve has a maximum point \(M\).

1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Show that the \(x\)-coordinate of \(M\) satisfies the equation \(x = \frac { x + 3 } { \ln ( 2 x + 1 ) } - 0.5\).
3. Show by calculation that the \(x\)-coordinate of \(M\) lies between 2.5 and 3.0 .
4. Use an iterative formula based on the equation in part (b) to find the \(x\)-coordinate of \(M\) correct to 4 significant figures. Give the result of each iteration to 6 significant figures.

5 The equation of a curve is \(y = \mathrm { e } ^ { - 2 x } \tan x\), for \(0 \leqslant x < \frac { 1 } { 2 } \pi\).

1. Obtain an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and show that it can be written in the form \(\mathrm { e } ^ { - 2 x } ( a + b \tan x ) ^ { 2 }\), where \(a\) and \(b\) are constants.
2. Explain why the gradient of the curve is never negative.
3. Find the value of \(x\) for which the gradient is least.

5 The equation of a curve is \(y = \mathrm { e } ^ { - 2 x } \tan x\), for \(0 \leqslant x < \frac { 1 } { 2 } \pi\).

1. Obtain an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and show that it can be written in the form \(\mathrm { e } ^ { - 2 x } ( a + b \tan x ) ^ { 2 }\), where \(a\) and \(b\) are constants.
2. Explain why the gradient of the curve is never negative.
3. Find the value of \(x\) for which the gradient is least.

7 The equation of a curve is \(y = \frac { \sin 2 x } { \cos x + 1 }\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 \left( \cos ^ { 2 } x + \cos x - 1 \right) } { \cos x + 1 }\).
2. Find the \(x\)-coordinate of each stationary point of the curve in the interval \(- \pi < x < \pi\). Give each answer correct to 3 significant figures.

4 The curve \(y = \mathrm { e } ^ { - 4 x } \tan x\) has two stationary points in the interval \(0 \leqslant x < \frac { 1 } { 2 } \pi\).

1. Obtain an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and show it can be written in the form \(\sec ^ { 2 } x ( a + b \sin 2 x ) \mathrm { e } ^ { - 4 x }\), where \(a\) and \(b\) are constants.
2. Hence find the exact \(x\)-coordinates of the two stationary points.

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 \(y = g ( x )\).

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 \(k\).

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. (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}

3. Given that $$y = \frac { \cos 2 \theta } { 1 + \sin 2 \theta } , \quad - \frac { \pi } { 4 } < \theta < \frac { 3 \pi } { 4 }$$ show that $$\frac { \mathrm { d } y } { \mathrm {~d} \theta } = \frac { a } { 1 + \sin 2 \theta } , \quad - \frac { \pi } { 4 } < \theta < \frac { 3 \pi } { 4 }$$ where \(a\) is a constant to be determined.

1. The curve \(C\) has equation

$$y = \frac { 3 + \sin 2 x } { 2 + \cos 2 x }$$

1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 6 \sin 2 x + 4 \cos 2 x + 2 } { ( 2 + \cos 2 x ) ^ { 2 } }$$
2. Find an equation of the tangent to \(C\) at the point on \(C\) where \(x = \frac { \pi } { 2 }\). Write your answer in the form \(y = a x + b\), where \(a\) and \(b\) are exact constants.

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 \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 8 } { ( x - 1 ) ^ { 3 } }\).

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 \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = - \frac { 15 } { 4 }\).

7. The function f is defined by $$\mathrm { f } ( x ) = 1 - \frac { 2 } { ( x + 4 ) } + \frac { x - 8 } { ( x - 2 ) ( x + 4 ) } , \quad x \in \mathbb { R } , x \neq - 4 , x \neq 2$$

1. Show that \(\mathrm { f } ( x ) = \frac { x - 3 } { x - 2 }\) The function g is defined by $$\mathrm { g } ( x ) = \frac { \mathrm { e } ^ { x } - 3 } { \mathrm { e } ^ { x } - 2 } , \quad x \in \mathbb { R } , x \neq \ln 2$$
2. Differentiate \(\mathrm { g } ( x )\) to show that \(\mathrm { g } ^ { \prime } ( x ) = \frac { \mathrm { e } ^ { x } } { \left( \mathrm { e } ^ { x } - 2 \right) ^ { 2 } }\)
3. Find the exact values of \(x\) for which \(\mathrm { g } ^ { \prime } ( x ) = 1\)

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 A curve has equation \(y = \frac { x \mathrm { e } ^ { 2 x } } { x + k }\), where \(k\) is a non-zero constant.

1. Differentiate \(x \mathrm { e } ^ { 2 x }\), and show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { \mathrm { e } ^ { 2 x } \left( 2 x ^ { 2 } + 2 k x + k \right) } { ( x + k ) ^ { 2 } }\).
2. Given that the curve has exactly one stationary point, find the value of \(k\), and determine the exact coordinates of the stationary point.

3

1. Differentiate \(\sqrt { 1 + 3 x ^ { 2 } }\).
2. Hence show that the derivative of \(x \sqrt { 1 + 3 x ^ { 2 } }\) is \(\frac { 1 + 6 x ^ { 2 } } { \sqrt { 1 + 3 x ^ { 2 } } }\).

8 You are given that \(\mathrm { f } ( x ) = \frac { x } { x ^ { 2 } + 1 }\) for all real values of \(x\).

1. Show that \(\mathrm { f } ^ { \prime } ( x ) = \frac { 1 - x ^ { 2 } } { \left( x ^ { 2 } + 1 \right) ^ { 2 } }\).
2. Hence show that there is a stationary value at \(\left( 1 , \frac { 1 } { 2 } \right)\) and find the coordinates of the other stationary point.
3. The graph of the curve is shown in Fig. 8. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{2f403099-2813-40d8-a9ae-1f7e64d41f80-3_518_892_1612_705} \captionsetup{labelformat=empty} \caption{Fig. 8}
   \end{figure} State whether the curve is odd or even and prove the result algebraically.
4. Show that \(\int _ { 1 } ^ { 4 } \frac { x } { x ^ { 2 } + 1 } \mathrm {~d} x = \int _ { a } ^ { b } k \frac { 1 } { u + 1 } \mathrm {~d} u\), where the values of \(a , b\) and \(k\) are to be determined.
5. Hence find the area of the shaded region in Fig. 8.

1

1. Differentiate \(\sqrt { 1 + 3 x ^ { 2 } }\).
2. Hence show that the derivative of \(x \sqrt { 1 + 3 x ^ { 2 } }\) is \(\frac { 1 + 6 x ^ { 2 } } { \sqrt { 1 + 3 x ^ { 2 } } }\).

7 Given that \(y = x ^ { 2 } \sqrt { 1 + 4 x }\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 x ( 5 x + 1 ) } { \sqrt { 1 + 4 x } }\).

2 Fig. 8 shows the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = \frac { x } { \sqrt { 2 + x ^ { 2 } } }\) \begin{figure}[h] \end{figure}

1. Show algebraically that \(\mathrm { f } ( x )\) is an odd function. Interpret this result geometrically.
2. Show that \(\mathrm { f } ^ { \prime } ( x ) = \frac { 2 } { \left( 2 + x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } }\). Hence find the exact gradient of the curve at the origin.
3. Find the exact area of the region bounded by the curve, the \(x\)-axis and the line \(x = 1\).
4. \(( A )\) Show that if \(y = \frac { x } { \sqrt { 2 + x ^ { 2 } } }\), then \(\frac { 1 } { y ^ { 2 } } = \frac { 2 } { x ^ { 2 } } + 1\).
   (B) Differentiate \(\frac { 1 } { y ^ { 2 } } = \frac { 2 } { x ^ { 2 } } + 1\) implicitly to show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 y ^ { 3 } } { x ^ { 3 } }\). Explain why this expression cannot be used to find the gradient of the curve at the origin.

8. \includegraphics[max width=\textwidth, alt={}, center]{a86f277c-a2ec-4ba0-ab08-575cad2a5e53-3_424_698_246_479} The diagram shows the curve \(y = \mathrm { f } ( x )\) in the interval \(0 \leq x \leq 2 \pi\) where $$\mathrm { f } ( x ) = \frac { \cos x } { 2 - \sin x } , \quad x \in \mathbb { R }$$

1. Show that \(\mathrm { f } ^ { \prime } ( x ) = \frac { 1 - 2 \sin x } { ( 2 - \sin x ) ^ { 2 } }\).
2. Find an equation for the tangent to the curve \(y = \mathrm { f } ( x )\) at the point where \(x = \pi\).
3. Find the minimum and maximum values of \(\mathrm { f } ( x )\) in the interval \(0 \leq x \leq 2 \pi\).
4. Explain why your answers to part (c) are the minimum and maximum values of \(\mathrm { f } ( x )\) for all real values of \(x\).

9 Fig. 9 shows the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = \frac { 2 x ^ { 2 } - 1 } { x ^ { 2 } + 1 }\) for the domain \(0 \leqslant x \leqslant 2\). \begin{figure}[h] \end{figure}

1. Show that \(\mathrm { f } ^ { \prime } ( x ) = \frac { 6 x } { \left( x ^ { 2 } + 1 \right) ^ { 2 } }\), and hence that \(\mathrm { f } ( x )\) is an increasing function for \(x > 0\).
2. Find the range of \(\mathrm { f } ( x )\).
3. Given that \(\mathrm { f } ^ { \prime \prime } ( x ) = \frac { 6 - 18 x ^ { 2 } } { \left( x ^ { 2 } + 1 \right) ^ { 3 } }\), find the maximum value of \(\mathrm { f } ^ { \prime } ( x )\). The function \(\mathrm { g } ( x )\) is the inverse function of \(\mathrm { f } ( x )\).
4. Write down the domain and range of \(\mathrm { g } ( x )\). Add a sketch of the curve \(y = \mathrm { g } ( x )\) to a copy of Fig. 9 .
5. Show that \(\mathrm { g } ( x ) = \sqrt { \frac { x + 1 } { 2 - x } }\).