Difference between revisions of "Derivative of coth"

From specialfunctionswiki
Jump to: navigation, search
 
(One intermediate revision by the same user not shown)
Line 1: Line 1:
<div class="toccolours mw-collapsible mw-collapsed">
+
==Theorem==
<strong>[[Derivative of coth|Theorem]]:</strong> The following formula holds:
+
The following formula holds:
 
$$\dfrac{\mathrm{d}}{\mathrm{d}z} \mathrm{coth}(z) = -\mathrm{csch}^2(z),$$
 
$$\dfrac{\mathrm{d}}{\mathrm{d}z} \mathrm{coth}(z) = -\mathrm{csch}^2(z),$$
 
where $\mathrm{coth}$ denotes the [[coth|hyperbolic cotangent]] and $\mathrm{csch}$ denotes the [[csch|hyperbolic cosecant]].
 
where $\mathrm{coth}$ denotes the [[coth|hyperbolic cotangent]] and $\mathrm{csch}$ denotes the [[csch|hyperbolic cosecant]].
<div class="mw-collapsible-content">
+
 
<strong>Proof:</strong> █
+
==Proof==
</div>
+
By the definition,
</div>
+
$$\mathrm{coth}(z) = \dfrac{\mathrm{cosh}(z)}{\mathrm{sinh}(z)}.$$
 +
Using the [[quotient rule]], the [[derivative of sinh]], the [[derivative of cosh]], [[Pythagorean identity for sinh and cosh]], and the definition of $\mathrm{csch}$, we see
 +
$$\begin{array}{ll}
 +
\dfrac{\mathrm{d}}{\mathrm{d}z} \mathrm{coth}(z) &= \dfrac{\sinh^2(z)-\cosh^2(z)}{\mathrm{sinh}^2(z)} \\
 +
&= - \dfrac{\mathrm{cosh}^2(z)-\mathrm{sinh}^2(z)}{\mathrm{sinh}^2(z)} \\
 +
&= -\mathrm{csch}^2(z),
 +
\end{array}$$
 +
as was to be shown.
 +
 
 +
==References==
 +
 
 +
[[Category:Theorem]]
 +
[[Category:Proven]]

Latest revision as of 12:18, 17 September 2016

Theorem

The following formula holds: $$\dfrac{\mathrm{d}}{\mathrm{d}z} \mathrm{coth}(z) = -\mathrm{csch}^2(z),$$ where $\mathrm{coth}$ denotes the hyperbolic cotangent and $\mathrm{csch}$ denotes the hyperbolic cosecant.

Proof

By the definition, $$\mathrm{coth}(z) = \dfrac{\mathrm{cosh}(z)}{\mathrm{sinh}(z)}.$$ Using the quotient rule, the derivative of sinh, the derivative of cosh, Pythagorean identity for sinh and cosh, and the definition of $\mathrm{csch}$, we see $$\begin{array}{ll} \dfrac{\mathrm{d}}{\mathrm{d}z} \mathrm{coth}(z) &= \dfrac{\sinh^2(z)-\cosh^2(z)}{\mathrm{sinh}^2(z)} \\ &= - \dfrac{\mathrm{cosh}^2(z)-\mathrm{sinh}^2(z)}{\mathrm{sinh}^2(z)} \\ &= -\mathrm{csch}^2(z), \end{array}$$ as was to be shown.

References