Derivative of Inverse Hyperbolic Functions

In this tutorial we shall 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 the formulae for the tangent inverse hyperbolic and cotangent inverse hyperbolic are the same.

Now for general formulas when any function is given in terms angles, they are of the following 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)