4.8 Implicit functions

1.

(a) $${\frac{dy}{dx} = \frac{1}{2y}, \frac{d^{2}y}{dx^{2}} = -\frac{1}{8y^{3}}}$$ =2.6zw =1(b) $${\frac{dy}{dx} = -\frac{2x+y}{x+4y}, \frac{d^{2}y}{dx^{2}} = \frac{-14(x^{2} + xy + 2y^{2})}{(x + 4y)^{3}}}$$

(c) $${\frac{dy}{dx} = e^{-y}, \frac{d^{2}y}{dx^{2}} = e^{-2y}}$$ (d) $${\frac{dy}{dx} = \frac{x^{2} - y}{x - y^{2}}, \frac{d^{2}y}{dx^{2}} = \frac{2xy(x^{3} - 3xy + y^{3} + 1)}{(x - y^{2})^{3}}}$$

2.

(a) $${\frac{dy}{dx} = \frac{z-x}{y-z}, \frac{dz}{dx} = \frac{x-y}{y-z}}$$ (b) $${\frac{dy}{dx} = \frac{y(z-x)}{x(y-z)}, \frac{dz}{dx} = \frac{z(x-y)}{x(y-z)}}$$