Calc Question With Tangent Lines

Question

Can someone help me with this calc problem:

$x^{\frac23}+y^{\frac23}=1$. Calculate the points where the tangent line has a slope of 1.

I know how to do it by finding the derivative but kept getting no solution, can anyone give me the right answer?

Answer 1

You should try implicit differentiation as follows: $\frac{d}{dx}(x^{\frac{2}{3}} + y^{\frac{2}{3}}) = \frac{2}{3}x^{-\frac{1}{3}} + \frac{2}{3}y^{-\frac{1}{3}}\frac{dy}{dx} = 0$.

Now express, $\frac{dy}{dx} = -x^{-\frac{1}{3}}y^{\frac{1}{3}}$ assuming $y\neq 0$. Then all points with slope of 1 will be precisely the ones in the set $\{(x,y)\in\mathbb{R}^2:-x^{-\frac{1}{3}}y^{\frac{1}{3}} = 1\}$.