Derivative Rules for Hyperbolic Functions

In this tutorial we shall discuss the 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 for general formulas, when any function is given in terms angles, they are of the following 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)