What is the equation of the tangent to the curve $$y = x^{1/3}$$ at the point $(0,0)$ ?
This is a homework question. I tried solving it. The derivative comes out to be infinite at the given point. So, the equation should be $x=0$. Am I doing it the right way?
On
Yes, correct. The tangent is $x=0$. This is because $$\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{1}{3x^{2/3}}$$ which gives us the form $\frac{1}{0}$ at $x=0$. This is an "undefined" or gradient of a straight line. Hence the tangent is $x=0$ or the $y$-axis as you (rightly) pointed out.
The above answer shows quite an elegant way of swapping the axes to see this is true, this answer is more of the normal computational plug and chug style.
Yes, that is correct. If you're uncomfortable with infinite derivatives, though, you can look at it as a function of $y$. You have $dx/dy=3y^2=0$ at $y=0$ so if you switch the axes, you get that the slope of the tangent is $0$ so the equation is $x=0$.