Higher-order partial differentiation

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Given the function \(z = f(x,y)\), suppose that the partial derivatives \(f_x(x,y)\) and \(f_y(x,y)\) exist, and they also have partial derivatives. For example, the partial derivative of \(f_x(x,y)\) with respect to \(y\),

\[(f_x)_y(x,y) = \frac{\partial}{\partial y}\left(\frac{\partial f}{\partial x}\right)(x,y) = \frac{\partial}{\partial y}\left(\frac{\partial z}{\partial x}\right)(x,y),\]

is denoted as \(\frac{\partial^2f}{\partial y\partial x}(x,y)\) or \(f_{xy}(x,y)\). Similarly, we may define \(f_{xx}(x,y)\), \(f_{yx}(x,y)\), \(f_{yy}(x,y)\). These are called second partial derivatives.

Example. Let \(f(x,y) = \log(x^2 + xy + 2y^2)\). Then,

\[\begin{eqnarray*} f_x(x,y) &=& \frac{2x + y}{x^2 + xy + 2y^2},\\ f_y(x,y) &=& \frac{x + 4y}{x^2 + xy + 2y^2},\\ f_{xx}(x,y) &=& -\frac{2x^2 + 2xy - 3y^2}{(x^2 + xy + 2y^2)^2},\\ f_{xy}(x,y) &=& -\frac{x^2 + 8xy + 2y^2}{(x^2 + xy + 2y^2)^2},\\ f_{yx}(x,y) &=& -\frac{x^2 + 8xy + 2y^2}{(x^2 + xy + 2y^2)^2},\\ f_{yy}(x,y) &=& \frac{3x^2 - 4xy - 8y^2}{(x^2 + xy + 2y^2)^2}. \end{eqnarray*}\] □

In the above example, we have \(f_{xy}(x,y) = f_{yx}(x,y)\). Is this generally the case? Short answer: Not always. In general, we have the following result.

Theorem (Changing the order of partial differentiation)

Suppose the function \(f(x,y)\) on an open region \(U\) has second derivatives \(f_{xy}(x,y)\) and \(f_{yx}(x,y)\) both of which are continuous. Then \(f_{xy}(x,y) = f_{yx}(x,y)\).

Proof. We show that \(f_{xy}(a,b) = f_{yx}(a,b)\) for every \((a, b) \in U\). 

Take a sufficiently small \(\delta > 0\) so that \(|x - a| < \delta\) and \(|y - b| < \delta\) imply \((x, y) \in U\) (Note: This is always possible because \(U\) is open). For all \(h,k\) such that \(0 < |h| < \delta\) and \(0 < |k| < \delta\), let

\[F(h,k) = f(a+h, b+k) - f(a+h, b) - f(a,b+k) + f(a,b).\]

Define the univariate function \(u(y)\) of \(y\) by

\[u(y) = f(a + h, y) - f(a,y).\]

Then we can write

\[F(h, k) = u(b+k) - u(b).\]

\(u(y)\) is differentiable with respect to \(y\) and \(u'(y) = f_y(a+h,y) - f_y(a,y)\). By the Mean Value Theorem, there exists \(\theta \in (0, 1)\) such that

\[F(h,k) = u'(b+\theta k)k = k[f_y(a+h, b+\theta k) - f_y(a, b+\theta k)].\]

  Next, consider the univariate function of \(x\): \(f_y(x, b + \theta k)\). Again, by the Mean Value Theorem, there exists \(\eta \in (0, 1)\) such that

\[F(h,k) = hkf_{yx}(a + \eta h, b + \theta k).\tag{Eq:Fhk1}\]

If we repeat the above process by swapping the roles of \(x\) and \(y\), we can show (exercise!) that there exist \(\theta', \eta' \in (0, 1)\) such that

\[F(h,k) = hkf_{xy}(a + \eta' h, b + \theta' k).\tag{Eq:Fhk2}\]

From Eqs. (Eq:Fhk1) and (Eq:Fhk2), it follows that

\[f_{yx}(a + \eta h, b + \theta k) = f_{xy}(a + \eta' h, b + \theta' k).\]

By the continuity of \(f_{xy}(x,y)\) and \(f_{yx}(x,y)\), as \((h,k) \to (0,0)\),

\[f_{yx}(a,b) = f_{xy}(a,b).\] ■

The second derivatives of the function \(f(x,y)\) may have derivatives. For example,

\[f_{yxy}(x,y) = \frac{\partial}{\partial y}f_{yx}(x,y).\]

If \(f_{xy}(x,y)\) and \(f_{yx}(x,y)\) are continuous, by the above theorem, \(f_{xy}(x,y) = f_{yx}(x,y)\). It follows that \(f_{xyy}(x,y) = f_{yxy}(x,y)\), that is,

\[\frac{\partial^3f}{\partial y^2\partial x} = \frac{\partial^3f}{\partial y\partial x\partial y}.\]

These are called third derivatives. 

Remark. We usually write

\[\frac{\partial^2 f}{\partial x^2}(x,y)\]

for \(f_{xx}(x,y)\), rather than

\[\frac{\partial^2 f}{\partial x\partial x}(x,y).\]

□

Definition (Functions of class \(C^n\))

Let \(f(x,y)\) be a function on an open region \(U\) and \(n\) be a non-negative integer.

  1. \(f(x,y)\) is said to be \(n\)-times continuously differentiable or of class \(C^n\) if it has all the partial derivatives up to the \(n\)-th order on \(U\), all of which are continuous.
  2. \(f(x,y)\) is said to be infinitely differentiable, smooth, or of class \(C^{\infty}\) if it has the partial derivatives of all orders on \(U\), all of which are continuous.
In general, the derivatives of a function of class \(C^n\) are determined by the number of partial differentiation with respect to \(x\) and \(y\) and are independent of the order of differentiation.

Example. If \(f(x,y)\) is of class \(C^3\), then \(f_{xxy}(x,y) = f_{xyx}(x,y) = f_{yxx}(x,y)\).



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