Derivative of Gudermannian

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, $$\dfrac{\mathrm{d}}{\mathrm{d}x} \mathrm{gd}(x) = \dfrac{1}{\cosh x} = \mathrm{sech}(x),$$ as was to be shown.