Practice question · True or false

For well-behaved (smooth) functions, the mixed partials agree: $f_{xy} = f_{yx}$ , differentiating in $x$ then $y$ gives the same result as $y$ then $x$ .

Hints

This is Clairaut’s theorem, check it on $f = x^{2}y$ if unsure.
Both routes on $x^{2}y$ give $2x$ .

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True

Why

True, for continuous second derivatives, the order of partial differentiation is interchangeable (Clairaut/Schwarz theorem). This hidden symmetry is why the Hessian matrix is symmetric and why "exactness" tests work for differential equations.

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Practise Partial Derivatives

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For well-behaved (smooth) functions, the mixed partials agree: fxy=fyxf_{xy} = f_{yx}fxy​=fyx​, differentiating in xxx then yyy gives the same result as yyy then xxx.

Practise Partial Derivatives

More questions on Partial Derivatives

For well-behaved (smooth) functions, the mixed partials agree: $f_{xy} = f_{yx}$ , differentiating in $x$ then $y$ gives the same result as $y$ then $x$ .