Calculus of a Single Variable, Chapter 5, 5.8, Section 5.8, Problem 27

f(x)=ln(sinh(x))
Take note that the derivative formula of natural logarithm is

d/dx[ln(u)]=1/u*(du)/dx
Applying this formula, the derivative of the function will be
f'(x)=d/dx[ln(sinh(x))]
f'(x)=1/(sinh(x))* d/dx[sinh(x)]
To take the derivative of hyperbolic sine, apply the formula

d/dx[sinh(u)] =cosh(u)*(du)/dx
So f'(x) will become
f'(x) =1/(sinh(x))* cosh(x)* d/dx(x)
f'(x)=1/(sinh(x))* cosh(x)*1
f'(x)= cosh(x)/sinh(x)
Since the ratio of hyperbolic cosine to hyperbolic sine is equal to hyperbolic cotangent, the f'(x) will simplify to
f'(x) =coth(x)

Therefore, the derivative of the given function is f'(x)=coth(x) .