Question

Encontre a derivada implícita de √xy=1+x² y

Answer 1

Resposta:

$y'=\frac{\frac{1}{2}(xy)^{-\frac{1}{2}}y-2xy}{x^2-(xy)^{-\frac{1}{2}}x}$

Explicação passo-a-passo:

$\sqrt{xy} =1+x^2y\\\frac{d}{dx}(xy)^{\frac{1}{2}}=\frac{d}{dx}(1+x^2y)\\\frac{1}{2}\frac{d}{dx}(xy)^{-\frac{1}{2}}(y+xy')=2xy+y'x^2 \\\frac{1}{2}\frac{d}{dx}(xy)^{-\frac{1}{2}}y+\frac{1}{2}(xy)^{-\frac{1}{2}}xy'=2xy+y'x^2\\ x^2y'-\frac{1}{2}(xy)}^{-\frac{1}{2}}}xy'=\frac{1}{2}(xy)^{-\frac{1}{2}}y-2xy\\y'=\frac{\frac{1}{2}(xy)^{-\frac{1}{2}}y-2xy}{x^2-(xy)^{-\frac{1}{2}}x}$