Chapter A6, Problem 22E

To determine

To find: the derivative of the given equation of y with respect to appropriate variable.

Answer to Problem 22E

The derivative is $\frac{dy}{dx}={\coth }^{3}x$ .

Explanation of Solution

Given:

  $y=\ln (\mathrm{sinhx})-\frac{1}{2}cot{h}^{2}x$

Calculation:

The derivative of y with respect to $x$ is calculated below

  $y=\ln (\mathrm{sinhx})-\frac{1}{2}cot{h}^{2}x$

Differentiating y with respect to $x$

  $\begin{array}{l}\frac{dy}{dx}=\frac{\cosh x}{\sinh x}-\frac{1}{2}(2\mathrm{cothx})(-csc{h}^{2}x)\\ \frac{dy}{dx}=\coth x+(cothx)(csc{h}^{2}x)\\ \frac{dy}{dx}=\coth x(1+{\mathrm{csch}}^{2}x)\\ \frac{dy}{dx}={\coth }^{3}x\end{array}$

Conclusion:

Therefore, the derivative is $\frac{dy}{dx}={\coth }^{3}x$ .