Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when:
\(\quad y = \ln ( 5 x - 2 )\);
\(y = \sin 2 x\).
The functions f and g are defined with their respective domains by
$$\begin{array} { l l }
\mathrm { f } ( x ) = \ln ( 5 x - 2 ) , & \text { for real values of } x \text { such that } x \geqslant \frac { 1 } { 2 }
\mathrm {~g} ( x ) = \sin 2 x , & \text { for real values of } x \text { in the interval } - \frac { \pi } { 4 } \leqslant x \leqslant \frac { \pi } { 4 }
\end{array}$$
Find the range of f .
Find an expression for \(\operatorname { gf } ( x )\).
Solve the equation \(\operatorname { gf } ( x ) = 0\).
The inverse of g is \(\mathrm { g } ^ { - 1 }\). Find \(\mathrm { g } ^ { - 1 } ( x )\).