# Derivative of Inverse Hyperbolic Functions

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

1. $\frac{d}{{dx}}{\sinh ^{ - 1}}x = \frac{1}{{\sqrt {1 + {x^2}} }}$

2. $\frac{d}{{dx}}{\cosh ^{ - 1}}x = \frac{1}{{\sqrt {{x^2} - 1} }}$

3. $\frac{d}{{dx}}{\tanh ^{ - 1}}x = \frac{1}{{1 - {x^2}}}$

4. $\frac{d}{{dx}}{\coth ^{ - 1}}x = \frac{1}{{1 - {x^2}}}$

5. $\frac{d}{{dx}}\sec {{\text{h}}^{ - 1}}x = - \frac{1}{{x\sqrt {1 - {x^2}} }}$

6. $\frac{d}{{dx}}\csc {{\text{h}}^{ - 1}}x = - \frac{1}{{x\sqrt {1 + {x^2}} }}$

It is observed that formulae tangent inverse hyperbolic and cotangent inverse hyperbolic are same.

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

1. $\frac{d}{{dx}}{\sinh ^{ - 1}}f\left( x \right) = \frac{1}{{\sqrt {1 + {{\left[ {f\left( x \right)} \right]}^2}} }}\frac{d}{{dx}}f\left( x \right)$

2. $\frac{d}{{dx}}{\cosh ^{ - 1}}f\left( x \right) = \frac{1}{{\sqrt {{{\left[ {f\left( x \right)} \right]}^2} - 1} }}\frac{d}{{dx}}f\left( x \right)$

3. $\frac{d}{{dx}}{\tanh ^{ - 1}}f\left( x \right) = \frac{1}{{1 - {{\left[ {f\left( x \right)} \right]}^2}}}\frac{d}{{dx}}f\left( x \right)$

4. $\frac{d}{{dx}}{\coth ^{ - 1}}f\left( x \right) = \frac{1}{{1 - {{\left[ {f\left( x \right)} \right]}^2}}}\frac{d}{{dx}}f\left( x \right)$

5. $\frac{d}{{dx}}\sec {{\text{h}}^{ - 1}}f\left( x \right) = - \frac{1}{{f\left( x \right)\sqrt {1 - {{\left[ {f\left( x \right)} \right]}^2}} }}\frac{d}{{dx}}f\left( x \right)$

6. $\frac{d}{{dx}}\csc {{\text{h}}^{ - 1}}f\left( x \right) = - \frac{1}{{f\left( x \right)\sqrt {1 + {{\left[ {f\left( x \right)} \right]}^2}} }}\frac{d}{{dx}}f\left( x \right)$