Higher Order Partial Derivatives — Question 2

Question 2

Let $g(x,y)=e^{xy}\cos y$.

Tasks

1. Compute $g_{xx}$ and $g_{xy}$.

2. Compute $g_{yx}$ independently from $g_y$.

3. Evaluate them at $(0,\pi)$ and verify agreement.

Question 2 – Solution

Strategy. Keep product-rule terms grouped by their exponential factor.

Step 1: Differentiate from $g_x$ Since $g_x=ye^{xy}\cos y$, $$\boxed{g_{xx}=y^2e^{xy}\cos y},$$ $$\boxed{g_{xy}=e^{xy}\bigl[(1+xy)\cos y-y\sin y\bigr]}.$$

Step 2: Reverse order $$g_y=e^{xy}(x\cos y-\sin y),$$ $$g_{yx}=ye^{xy}(x\cos y-\sin y)+e^{xy}\cos y =\boxed{e^{xy}[(1+xy)\cos y-y\sin y]}.$$

Step 3: Evaluate At $(0,\pi)$, $e^{xy}=1$, $\cos\pi=-1$, and $\sin\pi=0$: $$\boxed{g_{xx}=-\pi^2},\qquad\boxed{g_{xy}=g_{yx}=-1}.$$