- Find \(\frac { d y } { d x }\) for the following functions, simplifying your answers as far as possible.
i) \(y = \cos x - 2 \sin 2 x\)
ii) \(y = \frac { 1 } { 2 } x ^ { 4 } + 2 x ^ { 4 } \ln x\)
iii) \(y = \frac { 2 e ^ { 3 x } - 1 } { 3 e ^ { 3 x } - 1 }\)
a. Express \(\frac { 5 x + 7 } { ( x + 3 ) ( x + 1 ) ^ { 2 } }\) in partial fractions.
In this question you must show all of your algebraic steps clearly.
The function \(f ( x ) = \frac { 2 - 6 x + 5 x ^ { 2 } } { x ^ { 2 } ( 1 - 2 x ) }\) can be written in the form;
$$f ( x ) = \frac { - 2 } { x } + \frac { 2 } { x ^ { 2 } } + \frac { 1 } { 1 - 2 x }$$
b. Hence find the exact value of \(\int _ { 2 } ^ { 3 } \frac { 2 - 6 x + 5 x ^ { 2 } } { x ^ { 2 } ( 1 - 2 x ) } d x\)