Find the first partial derivatives of the function:

1) $\displaystyle f(x,y) = x^y$

2) $\displaystyle f(x,y) = \int_{y}^{x} cos(t^2) dt$

3) $\displaystyle u = x^{\frac{y}{z}}$

Solutions

1) $\displaystyle \frac{\partial f}{\partial x} = yx^{y-1}$ , $\displaystyle \frac{\partial f}{\partial y} = ln(x)$

2) I have no idea

3) $\displaystyle \frac{\partial u}{\partial x} = \frac{y}{z}x^{\frac{y}{z}-1}$ , $\displaystyle \frac{\partial u}{\partial y} = \frac{ln(x)}{z} = x^{\frac{1}{z}}$ , $\displaystyle \frac{\partial u}{\partial z} = \frac{yln(x)}{z^2} = x^{\frac{y}{z^2}}$

are these correct, and any idea on how to solve 2 would be greatly appreciated.