Differentiation of Simplified Fractions

1. The curve \(C\) has equation

$$y = \frac { \left( x ^ { 2 } + 4 \right) ( x - 3 ) } { 2 x } , \quad x \neq 0$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in its simplest form.
2. Find an equation of the tangent to \(C\) at the point where \(x = - 1\) Give your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

2. $$f ( x ) = \frac { 2 x + 2 } { x ^ { 2 } - 2 x - 3 } - \frac { x + 1 } { x - 3 }$$

1. Express \(\mathrm { f } ( x )\) as a single fraction in its simplest form.
2. Hence show that \(\mathrm { f } ^ { \prime } ( x ) = \frac { 2 } { ( x - 3 ) ^ { 2 } }\)

2. (a) Express $$\frac { 4 x - 1 } { 2 ( x - 1 ) } - \frac { 3 } { 2 ( x - 1 ) ( 2 x - 1 ) }$$ as a single fraction in its simplest form. Given that $$f ( x ) = \frac { 4 x - 1 } { 2 ( x - 1 ) } - \frac { 3 } { 2 ( x - 1 ) ( 2 x - 1 ) } - 2 , \quad x > 1$$ (b) show that $$f ( x ) = \frac { 3 } { 2 x - 1 }$$ (c) Hence differentiate \(\mathrm { f } ( x )\) and find \(\mathrm { f } ^ { \prime } ( 2 )\).

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.

9. Given that \(k\) is a negative constant and that the function \(\mathrm { f } ( x )\) is defined by $$f ( x ) = 2 - \frac { ( x - 5 k ) ( x - k ) } { x ^ { 2 } - 3 k x + 2 k ^ { 2 } } , \quad x \geqslant 0$$

1. show that \(\mathrm { f } ( x ) = \frac { x + k } { x - 2 k }\)
2. Hence find \(\mathrm { f } ^ { \prime } ( x )\), giving your answer in its simplest form.
3. State, with a reason, whether \(\mathrm { f } ( x )\) is an increasing or a decreasing function. Justify your answer.

$$f ( x ) = \frac { 2 x + 3 } { x + 2 } - \frac { 9 + 2 x } { 2 x ^ { 2 } + 3 x - 2 } , \quad x > \frac { 1 } { 2 }$$

1. Show that \(\mathrm { f } ( x ) = \frac { 4 x - 6 } { 2 x - 1 }\).
2. Hence, or otherwise, find \(\mathrm { f } ^ { \prime } ( x )\) in its simplest form.