Chain Rule — Question 10

Question 10 – Solution

Strategy. Differentiate $z'=f_xx'+f_yy'$ and apply the chain and product rules to every factor.

Step 1: General formula $$\boxed{z''=f_{xx}(x')^2+2f_{xy}x'y'+f_{yy}(y')^2+f_xx''+f_yy''}.$$ The first three terms measure the function’s second-order response along the velocity; the last two measure the effect of path acceleration on the first partials.

Step 2: Data For $f=x^2+xy+y^2$, $$f_x=2x+y,\ f_y=x+2y,\ f_{xx}=2,\ f_{xy}=1,\ f_{yy}=2.$$ At $t=0$, $(x,y)=(1,0)$, $(x',y')=(0,1)$, and $(x'',y'')=(-1,0)$.

Step 3: Evaluate $$z''=2(0)^2+2(1)(0)(1)+2(1)^2+(2)(-1)+(1)(0)=\boxed 0.$$ Directly, $z=\cos^2t+\sin t\cos t+\sin^2t=1+\tfrac 12\sin 2t$, whose second derivative is $-2\sin 2t$, also zero at $t=0$.