Given that \(\mathrm { y } = \frac { 4 \ln \mathrm { x } - 3 } { 4 \ln \mathrm { x } + 3 }\), show that \(\frac { \mathrm { dy } } { \mathrm { dx } } = \frac { 24 } { \mathrm { x } ( 4 \ln \mathrm { x } + 3 ) ^ { 2 } }\).
Find the exact value of the gradient of the curve \(y = \frac { 4 \ln x - 3 } { 4 \ln x + 3 }\) at the point where it crosses the \(x\)-axis.
\includegraphics[max width=\textwidth, alt={}, center]{133c38fb-307f-4f20-86cb-1bd57cc4f870-3_524_830_941_699}
The diagram shows part of the curve with equation
$$\mathrm { y } = \frac { 2 } { \mathrm { x } ^ { \frac { 1 } { 2 } } ( 4 \ln \mathrm { x } + 3 ) }$$
The region shaded in the diagram is bounded by the curve and the lines \(x = 1 , x = e\) and \(y = 0\). Find the exact volume of the solid produced when this shaded region is rotated completely about the x -axis.