# Derivative Rules for Hyperbolic Functions

In this tutorial we discuss basic formulas of differentiation for hyperbolic functions.

1. $\frac{d}{{dx}}\sinh x = \cosh x$

2. $\frac{d}{{dx}}\cosh x = \sinh x$

3. $\frac{d}{{dx}}\tanh x = \sec {{\text{h}}^2}x$

4. $\frac{d}{{dx}}\coth x = - \csc {{\text{h}}^2}x$

5. $\frac{d}{{dx}}\sec {\text{h}}x = - \sec {\text{h}}x\tanh x$

6. $\frac{d}{{dx}}\csc {\text{h}}x = - \csc {\text{h}}x\coth x$

Now general formulas when any function is given in terms angles, the following formula of the form

1. $\frac{d}{{dx}}\sinh f\left( x \right) = \cosh f\left( x \right)\frac{d}{{dx}}f\left( x \right)$

2. $\frac{d}{{dx}}\cosh f\left( x \right) = \sinh f\left( x \right)\frac{d}{{dx}}f\left( x \right)$

3. $\frac{d}{{dx}}\tanh f\left( x \right) = \sec {{\text{h}}^2}f\left( x \right)\frac{d}{{dx}}f\left( x \right)$

4. $\frac{d}{{dx}}\coth f\left( x \right) = - \csc {{\text{h}}^2}f\left( x \right)\frac{d}{{dx}}f\left( x \right)$

5. $\frac{d}{{dx}}\sec {\text{h}}f\left( x \right) = - \sec {\text{h}}f\left( x \right)\tanh f\left( x \right)\frac{d}{{dx}}f\left( x \right)$

6. $\frac{d}{{dx}}\csc {\text{h}}f\left( x \right) = - \csc {\text{h}}f\left( x \right)\coth f\left( x \right)\frac{d}{{dx}}f\left( x \right)$