Question

Calcule a derivada de F(X) = √(x/1+x^2)

Answer 1

regra da cadeia:
$\displaystyle \frac{d}{dx}\sqrt{\frac{x}{1+x^2}}\implies \frac{x}{1+x^2}=u\\\\ \frac{d}{dx}\sqrt{u}=\frac{d}{du}\sqrt{u}\cdot\frac{d}{dx}u=\frac{1}{2\sqrt{u}}\cdot\frac{1+x^2-2x^2}{(1+x^2)^2}=\frac{1}{2\sqrt{\frac{x}{1+x^2}}}\cdot\frac{1-x^2}{(1+x^2)^2}\\\\ \frac{d}{dx}\sqrt{\frac{x}{1+x^2}}=\boxed{\frac{1-x^2}{2\sqrt{\frac{x}{1+x^2}}\cdot(1+x^2)^2}}$