Problem

Suppose that y is defined implicitly as a function y(x) by an equation of the form...

Suppose that y is defined implicitly as a function y(x) by an equation of the form

F(x, y) = 0.

(For example, the equation x3 − y2 = 0 defines y as two functions of x, namely, y = x³^/² and y = −x³^/². The equation sin(xy) − x² y⁷ + ey = 0, on the other hand, cannot readily be solved for y in terms of x. See the end of §2.6 for more about implicit functions.) (a) Show that if F and y(x) are both assumed to be differentiable functions, then

provided Fy (x, y) 0. (b) Use the result of part (a) to find dy/dx when y is defined implicitly in terms of x by the equation x³ − y² = 0. Check your result by explicitly solving for y and differentiating.