Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2004 / Paper 1 / 9 / b

The hyperbolic cosine and hyperbolic sine are defined by $\boxed{\cosh x=(e^x+e^{-x})/2,\sinh x=(e^x-e^{-x})/2}$. Their quotient is $\tanh x=\sinh x/\cosh x$.

Expand the two squares using the exponential definitions:

The exponential definitions show that $\sinh$ is odd and $\cosh$ is even. Their quotient therefore satisfies $\boxed{\tanh (-x)=-\tanh x}$.

Differentiating the exponential definition gives $\boxed{(\cosh x{)}^{\prime }=(e^x-e^{-x})/2=\sinh x}$.