Question

Determine dy/dx for each of the following in terms of x and y, using implicit differentiation:
a) 3xy^2z+y^3=8

Answer 1

it's \(y = y(x)\), right??

but is it also \(z = z(x)\) or is z independent?

if we're Ok to treat z as a constant ie independent, there's a really handy shortcut for implicits like these, where \(f(x,y) = 3xy^2z+y^3=const\), as \(\frac{dy}{dx} = - \frac{f_x}{f_y} \)