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)