Higher Order Partial Derivatives — Question 1

Question 1 – Solution

Strategy. Compute both first partials before differentiating again.

Step 1: First partials $$f_x=4x^3y^2-6xy^3+2y,\qquad f_y=2x^4y-9x^2y^2+2x.$$

Step 2: Second partials $$\boxed{f_{xx}=12x^2y^2-6y^3},\qquad \boxed{f_{xy}=8x^3y-18xy^2+2},$$ $$\boxed{f_{yx}=8x^3y-18xy^2+2},\qquad \boxed{f_{yy}=2x^4-18x^2y}.$$

Step 3: Evaluate $$\boxed{f_{xx}(1,-1)=18},\quad \boxed{f_{xy}(1,-1)=f_{yx}(1,-1)=-24},\quad \boxed{f_{yy}(1,-1)=20}.$$ The two mixed formulas coincide term by term, as expected for a polynomial.