Question

Encontre a derivada da seguinte função inversa:

y = arccossec(x²).​​​​​​​​​​​​​​

Answer 1

Derivada da função de arco cossecante

$\boxed{\mathsf{\dfrac{d}{dx}arccsc(u)=\dfrac{-\frac{du}{dx}}{|u|.\sqrt{{u}^{2}-1}}}}$

$\mathsf{y=arccsc({x}^{2})}$

$\mathsf{\dfrac{dy}{dx}=\dfrac{-2x}{|{x}^{2}|\sqrt{{({x}^{2})}^{2}-1}}}$

$\mathsf{|{x}^{2}| > 0 \: \forall\,x\in~\mathbb{R} }$

$\mathsf{\dfrac{dy}{dx}=\dfrac{-2x}{{x}^{2}\sqrt{{({x}^{2})}^{2}-1}}}$

$\boxed{\boxed{\boxed{\mathsf{\dfrac{dy}{dx}=\dfrac{-2}{x\sqrt{{x}^{4}-1}}}}}}$