Find gradient at point

2 Find the gradient of the curve \(y = \frac { 12 } { x ^ { 2 } - 4 x }\) at the point where \(x = 3\).

2 A curve has equation \(y = 3 \tan \frac { 1 } { 2 } x \cos 2 x\).
Find the gradient of the curve at the point for which \(x = \frac { 1 } { 3 } \pi\).

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

1. \(y = x \ln ( x - 3 )\),
2. \(y = \frac { x - 1 } { x + 1 }\).

5 For each of the following curves, find the gradient at the point where the curve crosses the \(y\)-axis:

1. \(y = \frac { 1 + x ^ { 2 } } { 1 + \mathrm { e } ^ { 2 x } }\);
2. \(2 x ^ { 3 } + 5 x y + y ^ { 3 } = 8\).

2 The equation of a curve is \(y = \frac { \mathrm { e } ^ { 2 x } } { 1 + \mathrm { e } ^ { 2 x } }\). Show that the gradient of the curve at the point for which \(x = \ln 3\) is \(\frac { 9 } { 50 }\).

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 \(x\).

1. Given that

$$y = 3 x \arcsin 2 x \quad 0 \leqslant x \leqslant \frac { 1 } { 2 }$$

1. determine an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
2. Hence determine the exact value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(x = \frac { 1 } { 4 }\), giving your answer in the form \(a \pi + b\) where \(a\) and \(b\) are fully simplified constants to be found.

3

1. Differentiate \(x ^ { 2 } ( x + 1 ) ^ { 6 }\) with respect to \(x\).
2. Find the gradient of the curve \(y = \frac { x ^ { 2 } + 3 } { x ^ { 2 } - 3 }\) at the point where \(x = 1\).

4 Find the exact value of the gradient of the curve $$y = \sqrt { 4 x - 7 } + \frac { 4 x } { 2 x + 1 }$$ at the point for which \(x = 4\).

1

1. Use the mid-ordinate rule with four strips to find an estimate for \(\int _ { 1.5 } ^ { 5.5 } \mathrm { e } ^ { 2 - x } \ln ( 3 x - 2 ) \mathrm { d } x\), giving your answer to three decimal places.
   [0pt] [4 marks]
2. Find the exact value of the gradient of the curve \(y = \mathrm { e } ^ { 2 - x } \ln ( 3 x - 2 )\) at the point on the curve where \(x = 2\).
   [0pt] [4 marks]

For each of the following curves, find the gradient at the point with \(x\)-coordinate 2.

1. \(y = \frac{3x}{2x + 1}\) [3]
2. \(y = \sqrt{4x^2 + 9}\) [3]