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1.  Compute the gradient of the function \(f(x,y) = x e^{\frac{y}{x}}\)

a.  \( \nabla f(x,y) = \langle (ye^{\frac{y}{x}}, \frac{1}{x}e^{\frac{y}{x}}\rangle \)

b.  \( \nabla f(x,y) = \langle (1-\frac{y}{x})e^{\frac{y}{x}}, e^{\frac{y}{x}}\rangle \)

c.  \( \nabla f(x,y) = \langle (y e^{\frac{y}{x}}, x e^{\frac{y}{x}}\rangle \)

d.  \( \nabla f(x,y) = \langle (\frac{y}{x}e^{\frac{y}{x}}, x e^{\frac{y}{x}}\rangle \)

e.  \( \nabla f(x,y) = \langle (-\frac{x}{y})e^{\frac{y}{x}}, e^{\frac{y}{x}}\rangle \)

2.  If the equation \(\sin(xy) + \cos(yz) + xz = 4\) implicitly defines \(z\) as a function of \(x\) and \(y\), find \(\frac{\partial z}{\partial x}\).

a.  \(\frac{y\cos(xy)+z}{y\sin(yz)+x}\)

b.  \(-\frac{y\cos(xy)+z}{y\sin(yz)+x}\)

c.  \(\frac{(x+y)\cos(xy)+z}{(y+z)\sin(yz)+x}\)

d.  \(-\frac{(x+y)\cos(xy)+z}{(y+z)\sin(yz)+x}\)

e.  \(-\frac{(x+y)\cos(xy)+z}{(y+z)\sin(yz)+x+z}\)