Evaluate \( f_{x} \) and \( f_{y} \) at the given point. \[ f(x, y)=e^{12 y} \sin (x), \quad(\pi, 0) \]

To evaluate \( f_{x} \) and \( f_{y} \) for the function \( f(x, y) = e^{12y} \sin(x) \) at the point \( (\pi, 0) \), we first need to find the partial derivatives. 1. **Finding \( f_{x} \)**: The partial derivative of \( f \) with respect to \( x \) is calculated as follows: \[ f_{x} = \frac{\partial}{\partial x} (e^{12y} \sin(x)) = e^{12y} \cos(x) \] Now, substituting \( y = 0 \) into the derivative: \[ f_{x}(\pi, 0) = e^{12 \cdot 0} \cos(\pi) = 1 \cdot (-1) = -1 \] 2. **Finding \( f_{y} \)**: The partial derivative of \( f \) with respect to \( y \) is calculated as follows: \[ f_{y} = \frac{\partial}{\partial y} (e^{12y} \sin(x)) = 12 e^{12y} \sin(x) \] Now, substituting \( y = 0 \) into the derivative: \[ f_{y}(\pi, 0) = 12 e^{12 \cdot 0} \sin(\pi) = 12 \cdot 1 \cdot 0 = 0 \] In summary, the evaluations are: \[ f_{x}(\pi, 0) = -1, \quad f_{y}(\pi, 0) = 0 \]