Chain Rule — Question 2

Question 2 – Solution

Strategy. Evaluate the two chain-rule contributions before simplifying.

Step 1: Chain rule $$F_x=\frac{2x}{x^2+y^2},\qquad F_y=\frac{2y}{x^2+y^2},$$ $$x'=-3\sin t,\qquad y'=3\cos t.$$ Thus $$\frac{dF}{dt}=\frac{2x(-3\sin t)+2y(3\cos t)}{x^2+y^2}=0,$$ after substituting $x=3\cos t$, $y=3\sin t$.

Step 2: Meaning The curve remains on the circle $x^2+y^2=9$, and $F$ depends only on that squared radius. Hence $\boxed{dF/dt=0}$.

Step 3: Direct check $F(3\cos t,3\sin t)=\ln 9$, whose derivative is zero.