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Physics LibreTexts

59.2: Higher-Order Partial Derivatives

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It is similarly possible to take higher-order partial derivatives. For a function of two variables f(x,y), there are three possible second derivatives:

2fx2=x(fx);2fxy=x(fy); and 2fy2=y(fy).

In the second case, the order of differentiation doesn't matter: 2f/(xy)2f/(yx). This property is known as Clairaut's theorem.

For example, suppose f(x,y) is as given by Eq. 59.1.3. Then the second partial derivatives of f are found by taking partial derivatives of Eqs. 59.1.4 and 59.1.5:

2fx2=30xy52fxy=75x2y442y52fy2=100x3y3+8210xy4


59.2: Higher-Order Partial Derivatives is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by LibreTexts.

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