Differentiate logarithmic functions

3

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

2 Given that \(y = \ln ( 5 x )\)
find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
Circle your answer. $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { x } \quad \frac { \mathrm {~d} y } { \mathrm {~d} x } = \frac { 1 } { 5 x } \quad \frac { \mathrm {~d} y } { \mathrm {~d} x } = \frac { 5 } { x } \quad \frac { \mathrm {~d} y } { \mathrm {~d} x } = \ln 5$$