Partial Derivatives — Question 4

Question 4 – Solution

Strategy. Use the ordinary power rule for $x$ and write $x^y=e^{y\ln x}$ when differentiating with respect to the exponent.

Step 1: $x$-partial With $y$ fixed, $$\boxed{p_x(x,y)=yx^{y-1}}.$$

Step 2: $y$-partial Since $p=e^{y\ln x}$ and $\ln x$ is constant with respect to $y$, $$\boxed{p_y(x,y)=x^y\ln x}.$$

Step 3: Evaluate $$p_x(e,2)=2e,\qquad p_y(e,2)=e^2\ln e=e^2.$$ Thus $$\boxed{p_x(e,2)=2e,\qquad p_y(e,2)=e^2}.$$

Verification. For fixed $y=2$, $p=x^2$ has derivative $2x$. For fixed $x=e$, $p=e^y$ has derivative $e^y$, confirming both values.