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

Chapter 2, Problem 34

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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 x 3 y2 = 0 defines y as two functions of x, namely, y = x 3/2 and y = x 3/2. The equation sin(x y) x 2 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 dy dx = Fx (x, y) Fy (x, y) 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 x 3 y2 = 0. Check your result by explicitly solving for y and differentiating.

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