Find both first partial derivatives. \[ z=e^{x y} \]

To find the first partial derivatives of the function \( z = e^{xy} \), we start with \( \frac{\partial z}{\partial x} \) and \( \frac{\partial z}{\partial y} \). 1. For \( \frac{\partial z}{\partial x} \): We treat \( y \) as a constant, so we differentiate \( e^{xy} \) with respect to \( x \): \[ \frac{\partial z}{\partial x} = y e^{xy} \] 2. For \( \frac{\partial z}{\partial y} \): Similarly, we treat \( x \) as a constant and differentiate \( e^{xy} \) with respect to \( y \): \[ \frac{\partial z}{\partial y} = x e^{xy} \] Thus, the first partial derivatives are: \[ \frac{\partial z}{\partial x} = y e^{xy}, \quad \frac{\partial z}{\partial y} = x e^{xy} \]