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:
∂2f∂x2=∂∂x(∂f∂x);∂2f∂x∂y=∂∂x(∂f∂y); and ∂2f∂y2=∂∂y(∂f∂y).
In the second case, the order of differentiation doesn't matter: ∂2f/(∂x∂y)≡∂2f/(∂y∂x). 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:
∂2f∂x2=30xy5∂2f∂x∂y=75x2y4−42y5∂2f∂y2=100x3y3+8−210xy4