4.2 Partial derivatives

1.

(a) $${f_{x} = 6x - y, f_{y} = -x+1}$$ (b) $${f_{x} = 2xe^{-y}, f_{y} = -x^{2}e^{-y}}$$

(c) $${z_{x} = \frac{x}{\sqrt{x^{2} + y^{2}}}, z_{y} = \frac{y}{\sqrt{x^{2} + y^{2}}}}$$ (d) $${z_{x} = \sin{y}, z_{y} = x\cos{y}}$$

(e) $${z_{x} = \frac{2y}{(x+y)^{2}}, z_{y} = \frac{-2x}{x^{2} + y^{2}}}$$

2.

(a) $${f_{x} = 2ax + 2by, f_{y} =2bx + 2cy, f_{xx} = 2a, f_{xy} = 2b, f_{yy} = 2c}$$

=2.6zw =1(b) $${f_{x} = 2(x+y^{2} + z^{3}), f_{y} =4y(x+y^{2} + z^{3}), f_{z} = 6z^{2}(x + y^{2} + z^{3}),f_{xx} = 2}$$
$${ f_{xy} =4y, f_{yy} = 4(x + y^{2} + z^{3}), f_{yz} = 12yz^{2}, f_{zz} =12z(x+y^{2}) + 30z^{4}}$$

(c) $${f_{x} = 3\cos{(3x-2y)}, f_{y} = -2\cos(3x-2y), f_{xx} = -9\sin(3x-2y)}$$

$${f_{xy} = 6\sin(3x-2y), f_{yy} = 4\sin(3x-2y)}$$

(d) $${f_{x} = e^{2y}, f_{y} = 2xe^{2y}, f_{xx} = 0, f_{xy} = 2e^{2y}, f_{yy} = 4e^{2y}}$$