Higher Order Partial Derivatives — Question 7

Question 7 – Solution

Strategy. Follow the subscripts and keep polynomial and trigonometric contributions separate.

Step 1: $w_{xyz}$ $$w_x=2xyz^3+z\cos(xz),\quad w_{xy}=2xz^3,\quad \boxed{w_{xyz}=6xz^2}.$$

Step 2: $w_{zzx}$ $$w_z=3x^2yz^2+x\cos(xz),$$ $$w_{zz}=6x^2yz-x^2\sin(xz),$$ $$\boxed{w_{zzx}=12xyz-2x\sin(xz)-x^2z\cos(xz)}.$$

Step 3: Evaluate Both happen to equal $\boxed 0$ at $(1,2,0)$. Smoothness permits reordering derivatives that contain the same multiset of variables; $xyz$ and $zzx$ are different derivative types and need not agree generally.