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. $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { x } \quad \frac { \mathrm {~d} y } { \mathrm {~d} x } = \frac { 1 } { 5 x } \quad \frac { \mathrm {~d} y } { \mathrm {~d} x } = \frac { 5 } { x } \quad \frac { \mathrm {~d} y } { \mathrm {~d} x } = \ln 5$$

7

1. Differentiate \(3 \ln \left( x ^ { 2 } + 1 \right)\).
2. Find \(\int \frac { x ^ { 2 } } { 3 - 4 x ^ { 3 } } \mathrm {~d} x\).

Differentiate each of the following with respect to \(x\) and simplify your answers.

1. \(\ln(\cos x)\) [2]
2. \(x^2 \sin 3x\) [2]

Find \(\frac{dy}{dx}\) for the following functions, simplifying your answers as far as possible.

1. \(y = \cos x - 2 \sin 2x\) [2]
2. \(y = \frac{1}{2}x^4 + 2x^4 \ln x\) [3]
3. \(y = \frac{2e^{3x} - 1}{3e^{3x} - 1}\) [3]