Question

Determine the slope of x^(2/3)+y^(2/3)=5 at the point (8,1)

Answer 1

\[x^\frac{2}{3}+y^\frac{2}{3}=5\\
\frac{d}{dx}\left[x^\frac{2}{3}+y^\frac{2}{3}\right]=0\\
\frac{2}{3}x^{\frac{2}{3}-1}+\frac{2}{3}y^{\frac{2}{3}-1}\frac{dy}{dx}=0\]
Solving for dy/dx, you get an expression in terms of x and y. Plug in x = 8 and y = 1, and the resulting dy/dx is the slope you're looking for.