Find derivative of polynomial

1. Given that

$$y = 8 x ^ { 3 } - 4 \sqrt { } x + \frac { 3 x ^ { 2 } + 2 } { x } , \quad x > 0$$ find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
(6)

7. The point \(P ( 4 , - 1 )\) lies on the curve \(C\) with equation \(y = \mathrm { f } ( x ) , x > 0\), and $$f ^ { \prime } ( x ) = \frac { 1 } { 2 } x - \frac { 6 } { \sqrt { } x } + 3$$

1. Find the equation of the tangent to \(C\) at the point \(P\), giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are integers.
2. Find \(\mathrm { f } ( x )\).

Given \(y = x ^ { 3 } + 4 x + 1\), find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(x = 3\)

4. Given that \(y = 2 x ^ { 5 } + \frac { 6 } { \sqrt { } x } , x > 0\), find in their simplest form

1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
2. \(\int y \mathrm {~d} x\)

8. $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 x ^ { - \frac { 1 } { 2 } } + x \sqrt { } x , \quad x > 0$$ Given that \(y = 37\) at \(x = 4\), find \(y\) in terms of \(x\), giving each term in its simplest form.

11. \begin{figure}[h] \end{figure} A sketch of part of the curve \(C\) with equation $$y = 20 - 4 x - \frac { 18 } { x } , \quad x > 0$$ is shown in Figure 3. Point \(A\) lies on \(C\) and has an \(x\) coordinate equal to 2

1. Show that the equation of the normal to \(C\) at \(A\) is \(y = - 2 x + 7\) The normal to \(C\) at \(A\) meets \(C\) again at the point \(B\), as shown in Figure 3 .
2. Use algebra to find the coordinates of \(B\).

Given that \(y = 4 x ^ { 3 } - \frac { 5 } { x ^ { 2 } } , x \neq 0\), find in their simplest form

1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
2. \(\int y \mathrm {~d} x\)

1. Given that

$$y = 3 x ^ { 2 } + 6 x ^ { \frac { 1 } { 3 } } + \frac { 2 x ^ { 3 } - 7 } { 3 \sqrt { } x } , \quad x > 0$$ find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\). Give each term in your answer in its simplified form.

11. The curve \(C\) has equation \(y = 2 x ^ { 3 } + k x ^ { 2 } + 5 x + 6\), where \(k\) is a constant.

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) The point \(P\), where \(x = - 2\), lies on \(C\). The tangent to \(C\) at the point \(P\) is parallel to the line with equation \(2 y - 17 x - 1 = 0\)
   Find
2. the value of \(k\),
3. the value of the \(y\) coordinate of \(P\),
4. the equation of the tangent to \(C\) at \(P\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

2. Given $$y = \sqrt { x } + \frac { 4 } { \sqrt { x } } + 4 , \quad x > 0$$ find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(x = 8\), writing your answer in the form \(a \sqrt { 2 }\), where \(a\) is a rational number.
(5)

7. The curve \(C\) has equation \(y = \mathrm { f } ( x ) , x > 0\), where $$\mathrm { f } ^ { \prime } ( x ) = 30 + \frac { 6 - 5 x ^ { 2 } } { \sqrt { x } }$$ Given that the point \(P ( 4 , - 8 )\) lies on \(C\),

1. find the equation of the tangent to \(C\) at \(P\), giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants.
2. Find \(\mathrm { f } ( x )\), giving each term in its simplest form.

1. Given

$$y = 3 \sqrt { x } - 6 x + 4 , \quad x > 0$$

1. find \(\int y \mathrm {~d} x\), simplifying each term.
2. 1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
   2. Hence find the value of \(x\) such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\)

10. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of part of the curve \(C\) with equation $$y = \frac { 1 } { 2 } x + \frac { 27 } { x } - 12 , \quad x > 0$$ The point \(A\) lies on \(C\) and has coordinates \(\left( 3 , - \frac { 3 } { 2 } \right)\).

1. Show that the equation of the normal to \(C\) at \(A\) can be written as \(10 y = 4 x - 27\) The normal to \(C\) at \(A\) meets \(C\) again at the point \(B\), as shown in Figure 3.
2. Use algebra to find the coordinates of \(B\).

6. $$f ( x ) = \frac { ( 2 x + 1 ) ( x + 4 ) } { \sqrt { } x } , \quad x > 0$$

1. Show that \(\mathrm { f } ( x )\) can be written in the form \(P x ^ { \frac { 3 } { 2 } } + Q x ^ { \frac { 1 } { 2 } } + R x ^ { - \frac { 1 } { 2 } }\), stating the values of the constants \(P , Q\) and \(R\).
2. Find f \({ } ^ { \prime } ( x )\).
3. Show that the tangent to the curve with equation \(y = \mathrm { f } ( x )\) at the point where \(x = 1\) is parallel to the line with equation \(2 y = 11 x + 3\).
   (3)
   

   6. continued

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2. In this question you must show all stages of your working. \section*{Solutions relying entirely on calculator technology are not acceptable.} The curve \(C\) has equation $$y = 27 x ^ { \frac { 1 } { 2 } } - x ^ { \frac { 3 } { 2 } } - 20 \quad x > 0$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), giving each term in simplest form.
2. Hence find the coordinates of the stationary point of \(C\).
3. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) and hence determine the nature of the stationary point of \(C\).

7. The curve \(C\) has equation $$y = 2 x ^ { 3 } - 5 x ^ { 2 } - 4 x + 2$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Using the result from part (a), find the coordinates of the turning points of \(C\).
3. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
4. Hence, or otherwise, determine the nature of the turning points of \(C\).

3. $$y = x ^ { 2 } - k \sqrt { } x , \text { where } k \text { is a constant. }$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Given that \(y\) is decreasing at \(x = 4\), find the set of possible values of \(k\).

Given that \(y = 3 x ^ { 2 } + 4 \sqrt { } x , x > 0\), find

1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\),
3. \(\int y \mathrm {~d} x\).

Given that \(y = 2 x ^ { 5 } + 7 + \frac { 1 } { x ^ { 3 } } , x \neq 0\), find, in their simplest form, (a) \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
(b) \(\int y \mathrm {~d} x\).

7 Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in each of the following cases:

1. \(y = \frac { 1 } { 2 } x ^ { 4 } - 3 x\),
2. \(y = \left( 2 x ^ { 2 } + 3 \right) ( x + 1 )\),
3. \(y = \sqrt [ 5 ] { x }\).

7 Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in each of the following cases.

1. \(y = 5 x + 3\)
2. \(y = \frac { 2 } { x ^ { 2 } }\)
3. \(y = ( 2 x + 1 ) ( 5 x - 7 )\)

6 Given that \(\mathrm { f } ( x ) = ( x + 1 ) ^ { 2 } ( 3 x - 4 )\),

1. express \(\mathrm { f } ( x )\) in the form \(a x ^ { 3 } + b x ^ { 2 } + c x + d\),
2. find \(\mathrm { f } ^ { \prime } ( x )\),
3. find \(\mathrm { f } ^ { \prime \prime } ( x )\).

1 The points \(A ( 1,3 )\) and \(B ( 4,21 )\) lie on the curve \(y = x ^ { 2 } + x + 1\).

1. Find the gradient of the line \(A B\).
2. Find the gradient of the curve \(y = x ^ { 2 } + x + 1\) at the point where \(x = 3\).

7

1. Given that \(f ( x ) = x + \frac { 3 } { x }\), find \(f ^ { \prime } ( x )\).
2. Find the gradient of the curve \(\mathrm { y } = \mathrm { x } ^ { \frac { 5 } { 2 } }\) at the point where \(\mathrm { x } = 4\).