Difference between revisions of "Derivative of tanh"
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| − | + | ==Theorem== | |
| − | + | The following formula holds: | |
| − | + | $$\dfrac{\mathrm{d}}{\mathrm{d}z} \tanh(z)=\mathrm{sech}^2(z),$$ | |
| − | + | where $\tanh$ denotes the [[tanh|hyperbolic tangent]] and $\mathrm{sech}$ denotes the [[sech|hyperbolic secant]]. | |
| − | + | ||
| − | + | ==Proof== | |
| + | From the definition, | ||
| + | $$\tanh(z) = \dfrac{\sinh(z)}{\cosh(z)},$$ | ||
| + | and so using the [[derivative of sinh]], the [[derivative of cosh]], the [[quotient rule]], the [[Pythagorean identity for sinh and cosh]], and the definition of the [[sech|hyperbolic secant]], | ||
| + | $$\begin{array}{ll} | ||
| + | \dfrac{\mathrm{d}}{\mathrm{d}z} \tanh(z) &= \dfrac{\mathrm{d}}{\mathrm{d}z}\left[ \dfrac{\sinh(z)}{\cosh(z)} \right] \\ | ||
| + | &= \dfrac{\cosh^2(z)-\sinh^2(z)}{\cosh^2(z)} \\ | ||
| + | &= \dfrac{1}{\cosh^2(z)} \\ | ||
| + | &= \mathrm{sech}^2(z), | ||
| + | \end{array}$$ | ||
| + | as was to be shown. █ | ||
| + | |||
| + | ==References== | ||
| + | |||
| + | [[Category:Theorem]] | ||
| + | [[Category:Proven]] | ||
Latest revision as of 00:02, 17 June 2016
Theorem
The following formula holds: $$\dfrac{\mathrm{d}}{\mathrm{d}z} \tanh(z)=\mathrm{sech}^2(z),$$ where $\tanh$ denotes the hyperbolic tangent and $\mathrm{sech}$ denotes the hyperbolic secant.
Proof
From the definition, $$\tanh(z) = \dfrac{\sinh(z)}{\cosh(z)},$$ and so using the derivative of sinh, the derivative of cosh, the quotient rule, the Pythagorean identity for sinh and cosh, and the definition of the hyperbolic secant, $$\begin{array}{ll} \dfrac{\mathrm{d}}{\mathrm{d}z} \tanh(z) &= \dfrac{\mathrm{d}}{\mathrm{d}z}\left[ \dfrac{\sinh(z)}{\cosh(z)} \right] \\ &= \dfrac{\cosh^2(z)-\sinh^2(z)}{\cosh^2(z)} \\ &= \dfrac{1}{\cosh^2(z)} \\ &= \mathrm{sech}^2(z), \end{array}$$ as was to be shown. █
