Partial Derivatives — Question 5

Question 5 – Solution

Strategy. Differentiate with one variable active at a time and keep the logarithm’s domain restriction.

Step 1: Partial derivatives $$\boxed{F_x=2xyz+\frac 1{x+z}},$$ $$\boxed{F_y=x^2z-e^z},$$ $$\boxed{F_z=x^2y+\frac 1{x+z}-ye^z}.$$

Step 2: Evaluate At $(1,2,0)$, $$F_x=0+1=\boxed 1,\qquad F_y=0-1=\boxed{-1},$$ $$F_z=2+1-2=\boxed 1.$$

Verification. In $F_x$, $-ye^z$ is constant; in $F_y$, $\ln(x+z)$ is constant; in $F_z$, every term varies with $z$. The point satisfies $x+z=1>0$.