Question

The slope of the line tangent to the curve y^2+(xy+1)^3=0 at (2,-1) is what?

Answer 1

The slope of the tangent line is given by \(\dfrac{dy}{dx}\). Finding that requires implicit differentiation:
\[y^2+(xy+1)^3=0\]
Differentiate both sides with respect to \(x\).
\[2y\frac{dy}{dx}+3(xy+1)^2\left(y+x\frac{dy}{dx}\right)=0\]
Now just solve for \(\dfrac{dy}{dx}\), then plug in \(x=2\) and \(y=-1\).