Question

prove que a derivada de cossec é cosseccotg

Answer 1

$\mathrm{\dfrac{d}{dx}(\csc{x})=\dfrac{d}{dx}\bigg(\dfrac{1}{\sin{x}}\bigg)}$

$\mathrm{Regra\ do\ Reciproco\ \to\ \boxed{\mathrm{\bigg(\dfrac{1}{g(x)}\bigg)'=-\dfrac{g'(x)}{[g(x)]^2}}}}$

$\mathrm{\dfrac{d}{dx}\bigg(\dfrac{1}{\sin{x}}\bigg)=-\dfrac{(\sin{x})'}{(\sin{x})^2}=-\dfrac{\cos{x}}{\sin^2{x}}=\boxed{\mathbf{-\cot{x}\csc{x}}}}$