Difference between revisions of "Derivative of Gudermannian"
From specialfunctionswiki
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==Proof== | ==Proof== | ||
| + | From the definition, | ||
| + | $$\mathrm{gd}(x) = \displaystyle\int_0^x \dfrac{1}{\cosh t} \mathrm{d}t,$$ | ||
| + | where $\cosh$ denotes the [[cosh|hyperbolic cosine]]. Using the [[fundamental theorem of calculus]] and the definition of [[sech|hyperbolic secant]], | ||
| + | $$\dfrac{\mathrm{d}}{\mathrm{d}x} \mathrm{gd}(x) = \dfrac{1}{\cosh x} = \mathrm{sech}(x),$$ | ||
| + | as was to be shown. | ||
==References== | ==References== | ||
[[Category:Theorem]] | [[Category:Theorem]] | ||
| − | [[Category: | + | [[Category:Proven]] |
Revision as of 14:06, 19 September 2016
Theorem
The following formula holds: $$\dfrac{\mathrm{d}}{\mathrm{d}x} \mathrm{gd}(x)=\mathrm{sech}(x),$$ where $\mathrm{gd}$ denotes the Gudermannian and $\mathrm{sech}$ denotes the hyperbolic secant.
Proof
From the definition, $$\mathrm{gd}(x) = \displaystyle\int_0^x \dfrac{1}{\cosh t} \mathrm{d}t,$$ where $\cosh$ denotes the hyperbolic cosine. Using the fundamental theorem of calculus and the definition of hyperbolic secant, $$\dfrac{\mathrm{d}}{\mathrm{d}x} \mathrm{gd}(x) = \dfrac{1}{\cosh x} = \mathrm{sech}(x),$$ as was to be shown.
