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2023-03-30

What is the Mixed Derivative Theorem for mixed second-order partial derivatives? How can it help in calculating partial derivatives of second and higher orders?

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The Mixed Derivative Theorem states that for a function $f\left(x,y\right)$ with continuous second-order partial derivatives, the mixed partial derivatives $\frac{{\partial }^{2}f}{\partial y\partial x}$ and $\frac{{\partial }^{2}f}{\partial x\partial y}$ are equal.
Mathematically, the Mixed Derivative Theorem can be expressed as:
$\frac{{\partial }^{2}f}{\partial y\partial x}=\frac{{\partial }^{2}f}{\partial x\partial y}$
This theorem implies that the order in which we take the partial derivatives with respect to different variables does not matter, as long as the function has continuous second-order partial derivatives.
The Mixed Derivative Theorem is useful in calculating partial derivatives of second and higher orders because it allows us to simplify calculations and avoid redundancy. Instead of calculating both $\frac{{\partial }^{2}f}{\partial y\partial x}$ and $\frac{{\partial }^{2}f}{\partial x\partial y}$ separately, we only need to calculate one of them, as they are equal.
Here's an example to illustrate the concept:
Consider the function $f\left(x,y\right)=3{x}^{2}y+{y}^{3}$.
We can calculate the second-order partial derivatives using the Mixed Derivative Theorem:
First, we find the first-order partial derivatives:
$\frac{\partial f}{\partial x}=6xy$
$\frac{\partial f}{\partial y}=3{x}^{2}+3{y}^{2}$
Next, we calculate the second-order partial derivatives:
$\frac{{\partial }^{2}f}{\partial y\partial x}=\frac{\partial }{\partial y}\left(\frac{\partial f}{\partial x}\right)=\frac{\partial }{\partial y}\left(6xy\right)=6x$
$\frac{{\partial }^{2}f}{\partial x\partial y}=\frac{\partial }{\partial x}\left(\frac{\partial f}{\partial y}\right)=\frac{\partial }{\partial x}\left(3{x}^{2}+3{y}^{2}\right)=6x$
As we can see, the mixed partial derivatives $\frac{{\partial }^{2}f}{\partial y\partial x}$ and $\frac{{\partial }^{2}f}{\partial x\partial y}$ are equal, confirming the validity of the Mixed Derivative Theorem.
Thus, the Mixed Derivative Theorem simplifies the calculation of partial derivatives of second and higher orders by reducing the number of calculations required.

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