Partial Derivatives — Question 3

Question 3 – Solution

Strategy. Use the product rule, differentiating each composite factor with only one variable active.

Step 1: $x$-partial $$h_x=ye^{xy}\sin(x+y)+e^{xy}\cos(x+y) =e^{xy}\bigl[y\sin(x+y)+\cos(x+y)\bigr].$$

Step 2: $y$-partial $$h_y=xe^{xy}\sin(x+y)+e^{xy}\cos(x+y) =e^{xy}\bigl[x\sin(x+y)+\cos(x+y)\bigr].$$

Step 3: Evaluate At $(0,\pi/2)$, $e^{xy}=1$, $\sin(x+y)=1$, and $\cos(x+y)=0$. Therefore $$\boxed{h_x(0,\pi/2)=\pi/2},\qquad\boxed{h_y(0,\pi/2)=0}.$$ The cosine contribution vanishes in both partials; the $xe^{xy}\sin(x+y)$ contribution in $h_y$ also vanishes because $x=0$.