Find derivative after algebraic simplification (fractional/mixed powers)

3. A curve has equation $$y = \sqrt { 2 } x ^ { 2 } - 6 \sqrt { x } + 4 \sqrt { 2 } , \quad x > 0$$ Find the gradient of the curve at the point \(P ( 2,2 \sqrt { 2 } )\).
Write your answer in the form \(a \sqrt { 2 }\), where \(a\) is a constant.
(Solutions based entirely on graphical or numerical methods are not acceptable.) \(L\)

4. Given that $$y = 5 x ^ { 2 } + \frac { 1 } { 2 x } + \frac { 2 x ^ { 4 } - 8 } { 5 \sqrt { x } } \quad x > 0$$ find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), giving each term in its simplest form.
(6)

1. Given that

$$y = 4 x ^ { 3 } - 1 + 2 x ^ { \frac { 1 } { 2 } } , \quad x > 0 ,$$ find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\). \includegraphics[max width=\textwidth, alt={}, center]{fff086fd-f5d8-45b7-8db1-8b22ba5aab31-02_29_45_2690_1852}

5. (a) Write \(\frac { 2 \sqrt { } x + 3 } { x }\) in the form \(2 x ^ { p } + 3 x ^ { q }\) where \(p\) and \(q\) are constants. Given that \(y = 5 x - 7 + \frac { 2 \sqrt { } x + 3 } { x } , \quad x > 0\),
(b) find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), simplifying the coefficient of each term.

Given that \(y = x ^ { 4 } + 6 x ^ { \frac { 1 } { 2 } }\), find in their simplest form

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

5. Differentiate with respect to \(x\)

1. \(x ^ { 4 } + 6 \sqrt { } x\),
2. \(\frac { ( x + 4 ) ^ { 2 } } { x }\).

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)

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.

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.

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 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. \(y = ( 2 x + 1 ) ( 5 x - 7 )\)

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 \(\mathrm { x } = 4\).

5 Find the gradient of the curve \(y = 8 \sqrt { x } + x\) at the point whose \(x\)-coordinate is 9 .

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

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

1. Differentiate with respect to \(x\)

$$3 x ^ { 2 } - \sqrt { x } + \frac { 1 } { 2 x }$$

1 Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(y = x ^ { 6 } + \sqrt { x }\).

1 Differentiate \(6 x ^ { \frac { 5 } { 2 } } + 4\).

7. Differentiate with respect to \(x\), giving each answer in its simplest form.

1. \(( 1 - 2 x ) ^ { 2 }\)
2. \(\frac { x ^ { 5 } + 6 \sqrt { } x } { 2 x ^ { 2 } }\)

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

1. \(y = 10 x ^ { - 5 }\),
2. \(y = \sqrt [ 4 ] { x }\),
3. \(y = x ( x + 3 ) ( 1 - 5 x )\).

6 Find the gradient of the curve \(y = 2 x + \frac { 6 } { \sqrt { x } }\) at the point where \(x = 4\).

1. The curve with equation \(y = \mathrm { f } ( x )\) passes through the point (8, 7).

Given that $$\mathrm { f } ^ { \prime } ( x ) = 4 x ^ { \frac { 1 } { 3 } } - 5$$ find \(\mathrm { f } ( x )\).

Differentiate with respect to \(x\)

1. \(x^4 + 6\sqrt{x}\), [3]
2. \(\frac{(x + 4)^3}{x}\). [4]