In this entry we compute the derivative of the hyperbolic functions $\sinh(x)$ and $\cosh(x)$.

Recall that by definition:

Therefore:

Similarly $\displaystyle\frac{d}{dx}\cosh(x)=\sinh(x)$. Using the quotient rule, we compute the derivative of $\displaystyle\tanh(x)=\frac{\sinh(x)}{\cosh(x)}$:

where we have used the equality $\cosh^{2}(x)-\sinh^{2}(x)=1$.