question:$\displaystyle z=100x^{.25}y^{.75}$ where $\displaystyle x=48-4y$ Use chain rule to find $\displaystyle \frac{dz}{dy}$. Solve $\displaystyle \frac{dz}{dy}=0$ for y in terms of x and use the constraint to find x and y to maximize z.

NOTE: these are normal derivatives, not partial

work:

$\displaystyle z=100(48-4y)^{.25}y^{.75}$

I take the derivative and subsitute x back in to get $\displaystyle \frac{dz}{dy}=-x^{-.75}y^{.75}+\frac{3}{4}y^{-.25}x^{.25}$

Then I set it equal to 0 to get $\displaystyle \frac{3}{4}y^{-.25}x^{.25}=x^{-.75}y^{.75}$