Question

Use implicit differentiation to justify the vertical asymptote (x = -3) for the curve

Answer 1

The derivative of the y function represents the slope of the function on each point. y is a function of x, if we differentiate our curve, we're going to have

`\begin{gathered} (d)/(dx)(y^2)=(d)/(dx)(x^3+3x^2) \\ 2y(dy)/(dx)=3x^2+6x \\ (dy)/(dx)=(3x^2+6x)/(2y) \end{gathered}`

The derivative at x = -3 diverges(when x = -3, y = 0), therefore, we have a vertical asymptote.