Higher Order Partial Derivatives — Question 4

Question 4 – Solution

Strategy. Follow each subscript successively and retain all product-rule terms.

Step 1: Route $x,x,y$ $$f_x=e^{xy}(2x+x^2y),$$ $$f_{xx}=e^{xy}(2+4xy+x^2y^2),$$ $$\boxed{f_{xxy}=e^{xy}(6x+6x^2y+x^3y^2)}.$$

Step 2: Route $y,x,x$ $$f_y=x^3e^{xy},\qquad f_{yx}=e^{xy}(3x^2+x^3y),$$ $$\boxed{f_{yxx}=e^{xy}(6x+6x^2y+x^3y^2)}.$$

Step 3: Verify Polynomial and exponential factors are smooth, so the derivative order may be permuted. At $(1,0)$ both expressions equal $\boxed 6$.