Partial Derivatives — Question 6

Question 6 – Solution

Strategy. Differentiate the equation while treating $z=z(x,y)$, then collect the terms containing the requested derivative.

Step 1: Differentiate in $x$ $$2x+2zz_x+yz+xyz_x=0,$$ so $$\boxed{z_x=-\frac{2x+yz}{2z+xy}}.$$

Step 2: Differentiate in $y$ $$2y+2zz_y+xz+xyz_y=0,$$ so $$\boxed{z_y=-\frac{2y+xz}{2z+xy}}.$$

Step 3: Evaluate At $(1,1,1)$, the common denominator is $2z+xy=3\ne 0$. Hence $$\boxed{z_x(1,1)=-1,\qquad z_y(1,1)=-1}.$$ Substitution into the differentiated equations gives $2+2(-1)+1+(-1)=0$ in each case.