Question

To find the derivative of the function\(f(t) = cscht(1 - lncscht)\).

Short answer

The derivative of the function \(f(t) = {\mathop{\rm csch}\nolimits} t(1 - \ln {\mathop{\rm csch}\nolimits} t)\) is \((\ln {\mathop{\rm csch}\nolimits} t)({\mathop{\rm csch}\nolimits} t\coth t)\).

Step-by-step solution

Given function

The derivative of the function is \(f(t) = {\mathop{\rm csch}\nolimits} t(1 - \ln {\mathop{\rm csch}\nolimits} t)\).

Concept of derivative rules

The derivative of cosecant function is given by\(\frac{d}{{dx}}(cschx) = - cschxcothx\).

 Evaluate the derivative  

Derivation of \(f(t) = {\mathop{\rm csch}\nolimits} t(1 - \ln {\mathop{\rm csch}\nolimits} t)\).

\(\begin{aligned}{c}{f^\prime }(t) &= {({\mathop{\rm csch}\nolimits} t(1 - \ln {\mathop{\rm csch}\nolimits} t))^\prime }\\ &= {\mathop{\rm csch}\nolimits} t\left( { - \frac{1}{{{\mathop{\rm csch}\nolimits} t}}} \right)( - {\mathop{\rm csch}\nolimits} t\coth t) + (1 - \ln {\mathop{\rm csch}\nolimits} t)( - {\mathop{\rm csch}\nolimits} t\coth t)\\ &= {\mathop{\rm csch}\nolimits} t\coth t - {\mathop{\rm csch}\nolimits} t\coth t + (\ln {\mathop{\rm csch}\nolimits} t)({\mathop{\rm csch}\nolimits} t\coth t)\\ &= (\ln {\mathop{\rm csch}\nolimits} t)({\mathop{\rm csch}\nolimits} t\coth t)\end{aligned}\)

Therefore, the derivative of the function \(f(t) = {\mathop{\rm csch}\nolimits} t(1 - \ln {\mathop{\rm csch}\nolimits} t)\) is \((\ln {\mathop{\rm csch}\nolimits} t)({\mathop{\rm csch}\nolimits} t\coth t).\)