Difference between revisions of "Derivative of hyperbolic cosecant"
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| − | + | ==Theorem== | |
| − | + | The following formula holds: | |
$$\dfrac{\mathrm{d}}{\mathrm{d}z} \mathrm{csch}(z)=-\mathrm{csch}(z)\mathrm{coth}(z),$$ | $$\dfrac{\mathrm{d}}{\mathrm{d}z} \mathrm{csch}(z)=-\mathrm{csch}(z)\mathrm{coth}(z),$$ | ||
where $\mathrm{csch}$ denotes the [[csch|hyperbolic cosecant]] and $\mathrm{coth}$ denotes the [[coth|hyperbolic cotangent]]. | where $\mathrm{csch}$ denotes the [[csch|hyperbolic cosecant]] and $\mathrm{coth}$ denotes the [[coth|hyperbolic cotangent]]. | ||
| − | + | ||
| − | + | ==Proof== | |
| − | + | From the definition, | |
| − | + | $$\mathrm{csch}(z) = \dfrac{1}{\mathrm{sinh}(z)}.$$ | |
| + | Using the [[quotient rule]], the [[derivative of sinh]], and the definition of $\mathrm{coth}$, we compute | ||
| + | $$\begin{array}{ll} | ||
| + | \dfrac{\mathrm{d}}{\mathrm{d}z} \mathrm{csch}(z) &= \dfrac{0-\mathrm{cosh}(z)}{\mathrm{sinh}^2(z)} \\ | ||
| + | &= -\mathrm{csch}(z)\mathrm{coth}(z), | ||
| + | \end{array}$$ | ||
| + | as was to be shown. | ||
| + | ==References== | ||
| + | |||
| + | [[Category:Theorem]] | ||
| + | [[Category:Proven]] | ||
Latest revision as of 12:13, 17 September 2016
Theorem
The following formula holds: $$\dfrac{\mathrm{d}}{\mathrm{d}z} \mathrm{csch}(z)=-\mathrm{csch}(z)\mathrm{coth}(z),$$ where $\mathrm{csch}$ denotes the hyperbolic cosecant and $\mathrm{coth}$ denotes the hyperbolic cotangent.
Proof
From the definition, $$\mathrm{csch}(z) = \dfrac{1}{\mathrm{sinh}(z)}.$$ Using the quotient rule, the derivative of sinh, and the definition of $\mathrm{coth}$, we compute $$\begin{array}{ll} \dfrac{\mathrm{d}}{\mathrm{d}z} \mathrm{csch}(z) &= \dfrac{0-\mathrm{cosh}(z)}{\mathrm{sinh}^2(z)} \\ &= -\mathrm{csch}(z)\mathrm{coth}(z), \end{array}$$ as was to be shown.
