First we can isolate y:
[tex]\begin{gathered} x^2+xy=10 \\ xy=10-x^2 \\ y=\frac{10}{x}-x \\ y=10x^{-1}-x \end{gathered}[/tex]
To derive that, we can do it term by term, using the power rule:
[tex]\frac{d}{dx}(x^n)=nx^{n-1}[/tex]
So:
[tex]\frac{dy}{dx}=10(-1)x^{-1-1}-(1)x^{1-1}=-10x^{-2}-1^{}[/tex]
Now we substitute x = 2:
[tex]\frac{dy}{dx}(x=2)=-10(2)^{-2}-1=-\frac{10}{4}-1=-\frac{5}{2}-\frac{2}{2}=-\frac{7}{2}[/tex]
So the answer is alternative A.
