Question

Find the derivative. Simplify where possible.

40.\(F\left( t \right) = \ln \left( {\sinh t} \right)\)

Short answer

The derivative is \(F'\left( t \right) = \coth t\).

Step-by-step solution

Apply the derivative of composite functions

On applying the derivative to the given function\(F\left( t \right) = \ln \left( {\sinh t} \right)\), we get, as follows:

^.\(F'\left( t \right) = \frac{d}{{dt}}\ln \left( {\sinh t} \right)\frac{d}{{dt}}\left( {\sinh t} \right)\)

Plug in the derivatives and simplify

The derivative of\(\ln \left( {\sinh t} \right)\)is \(\frac{1}{{\sinh t}}\) and that of \(\sinh t\)is \(\cosh t\). On plugging this derivative and simplifying we get:

\(\begin{aligned}F'\left( t \right) &= \left( {\frac{1}{{\sinh t}}} \right)\left( {\cosh t} \right)\\ &= \frac{{\cosh t}}{{\sinh t}}\\ &= \coth t\end{aligned}\)

So, \(F'\left( t \right) = \coth t\).