Implicit functions

Exercise

1.: Find $\displaystyle{\frac{dy}{dx}, \ \frac{d^{2}y}{dx^{2}}}$ determined by the following function ．
(a): $\displaystyle{2x^2+ 5xy - 3y^2 = 1}$
(b): $\displaystyle{y = e^{x+y}}$
(c): $\displaystyle{x^2 - y^2 = xy}$
(d): $\displaystyle{\log{\sqrt{x^2 + y^2}} = \tan^{-1}{\frac{y}{x}}}$
2.: Find $\displaystyle{\frac{dy}{dx}, \ \frac{dz}{dx}}$ determined by the following function．
(a): $\displaystyle{x^2 +y^2 + z^2 = 4, x^2 + y^2 = 4x}$
(b): $\displaystyle{xyz = 1, xy + yz + zx = 1}$
3.: Find the tangent line and the normla line of the curve at $(1,\sqrt{2})$ ．
4.: Find the tangent plane and normal line of the surface $\displaystyle{z = \tan^{-1}\frac{y}{x}}$ at $(1,1,\frac{\pi}{2})$ ．