Answer any two questions from this section. 1. a) \( \begin{array}{l}\text { If } f(x, y)=\sin x+e^{x y} \text {, then find } \\ \text { (i) } \nabla f(x, y)\end{array} \) [5 Marks] [3 Marks] \( 8 f(0,1) \)

To find the gradient \( \nabla f(x, y) \) of the function \( f(x, y) = \sin x + e^{xy} \), we need to compute the partial derivatives with respect to \( x \) and \( y \). 1. The partial derivative with respect to \( x \): \[ \frac{\partial f}{\partial x} = \cos x + y e^{xy} \] 2. The partial derivative with respect to \( y \): \[ \frac{\partial f}{\partial y} = x e^{xy} \] Thus, the gradient is: \[ \nabla f(x, y) = \left( \cos x + y e^{xy}, x e^{xy} \right) \] Now, we can evaluate \( 8 f(0, 1) \): \[ f(0, 1) = \sin(0) + e^{0 \cdot 1} = 0 + e^0 = 1 \] Thus, \[ 8 f(0, 1) = 8 \cdot 1 = 8 \]