Exercise 1083Calculus › Multivariable calculus › Partial derivativesL1Added 3 Nov 2026

Second-order partial derivatives

#higher order#Schwarz#second derivatives

Statement

Let $f(x,y)=x^3y^2+xy$ . Compute all four second-order partial derivatives and check Schwarz's theorem.

Hint

Differentiate $f_x$ and $f_y$ once more; compare $f_{xy}$ and $f_{yx}$ .

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First derivatives. $f_x=3x^2y^2+y$ , $f_y=2x^3y+x$ .

Second derivatives. $f_{xx}=6xy^2$ , $f_{yy}=2x^3$ , $f_{xy}=6x^2y+1$ , $f_{yx}=6x^2y+1$ .

Conclusion. $f_{xy}=f_{yx}$ : Schwarz's theorem holds (the mixed partials are equal because $f$ is $C^2$ ).