Question

calcule a derivada da função f(x)=tgh^-1(sen x)

Answer 1

$f(x)=arctanh(\sin x)\\\\ \displaystyle \frac{d}{dx}arc\tanh(\sin x)=\frac{d}{du}arctanh(u)\cdot\frac{d}{dx}\sin x$
analisando:
$\displaystyle \tanh(x)=\frac{1-e^{-2x}}{1+e^{-2x}}\\\\\frac{d}{dx}\tanh(x)=\frac{1}{\cosh^2(x)}\\ \tanh^{-1}(x)=arctanh(x)\\\\ \frac{d}{dx}arctanh(x)=\frac{1}{1-x^2}$
então:
$\displaystyle \frac{d}{dx}arctanh(\sin x)=\frac{d}{du}arctanh(u)\cdot\frac{d}{dx}\sin x\implies \\\\ \frac{d}{dx}arctanh(\sin x)=\frac{1}{1-u^2}\cdot -\cos x=\boxed{-\frac{\cos x}{1-\sin^2x}}$