• Algebra
• Trigonometry
• Geometry
• Calculus

Derivative of Hyperbolic Cotangent function

Formula

$\dfrac{d}{dx}{\, \coth{x}} \,=\, -\operatorname{csch}^2{x}$

Introduction

When a variable is denoted by $x$, the hyperbolic cotangent function is written as $\coth{(x)}$ in mathematics. The derivative of the hyperbolic function with respect to $x$ is written in the following mathematical form in differential calculus.

$\dfrac{d}{dx}{\, \coth{x}}$

The derivative of the hyperbolic cotangent function is equal to the negative square of hyperbolic co-secant function.

$\implies$ $\dfrac{d}{dx}{\, \coth{(x)}} \,=\, -\operatorname{csch}^2{(x)}$

Alternative forms

The derivative rule of hyperbolic cotangent function can also be written in terms of any variable.

$(1) \,\,\,$ $\dfrac{d}{dl}{\, \coth{(l)}} \,=\, -\operatorname{csch}^2{(l)}$

$(2) \,\,\,$ $\dfrac{d}{dp}{\, \coth{(p)}} \,=\, -\operatorname{csch}^2{(p)}$

$(3) \,\,\,$ $\dfrac{d}{dz}{\, \coth{(z)}} \,=\, -\operatorname{csch}^2{(z)}$

Proof

Learn how to prove the derivative formula of the hyperbolic cotangent function from the first principle of the differentiation.