Find the partial derivative of (5 Marks) \( f(x, y)=e^{-t x} \cos (\pi y) \) Where \( t \) is a constant. a) \( \frac{\partial f}{\partial x}= \) b) \( \frac{\partial f}{\partial y}=\square \) Hint: use pi in place of \( \pi \) in the answer

To find the partial derivatives of the function \( f(x, y) = e^{-t x} \cos(\pi y) \), we'll differentiate with respect to each variable while treating the other variable as a constant. ### a) Partial derivative with respect to \( x \): \[ \frac{\partial f}{\partial x} = \frac{\partial}{\partial x} \left( e^{-t x} \cos(\pi y) \right) \] Since \( \cos(\pi y) \) is treated as a constant when differentiating with respect to \( x \): \[ \frac{\partial f}{\partial x} = -t e^{-t x} \cos(\pi y) \] ### b) Partial derivative with respect to \( y \): \[ \frac{\partial f}{\partial y} = \frac{\partial}{\partial y} \left( e^{-t x} \cos(\pi y) \right) \] Here, \( e^{-t x} \) is treated as a constant when differentiating with respect to \( y \): \[ \frac{\partial f}{\partial y} = -e^{-t x} \pi \sin(\pi y) \] **Final Answers:** - **a)** \( \frac{\partial f}{\partial x} = -t\, e^{-t x} \cos(\pi y) \) - **b)** \( \frac{\partial f}{\partial y} = -e^{-t x} \pi \sin(\pi y) \)