Difference between revisions of "Derivative of Gudermannian"
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==Proof== | ==Proof== | ||
| + | From the definition, | ||
| + | $$\mathrm{gd}(x) = \displaystyle\int_0^x \mathrm{sech}(t) \mathrm{d}t.$$ | ||
| + | Using the [[fundamental theorem of calculus]], | ||
| + | $$\dfrac{\mathrm{d}}{\mathrm{d}x} \mathrm{gd}(x) = \mathrm{sech}(x),$$ | ||
| + | as was to be shown. | ||
==References== | ==References== | ||
| + | |||
| + | [[Category:Theorem]] | ||
| + | [[Category:Proven]] | ||
Latest revision as of 22:08, 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, $$\mathrm{gd}(x) = \displaystyle\int_0^x \mathrm{sech}(t) \mathrm{d}t.$$ Using the fundamental theorem of calculus, $$\dfrac{\mathrm{d}}{\mathrm{d}x} \mathrm{gd}(x) = \mathrm{sech}(x),$$ as was to be shown.
