1.3 Partial Derivative

The ultimate goal of calculus is deriving the derivative and antiderivative of a given function. If so, what is the concept in multivariable calculus corresponding with "derivatives"? Yet we cannot propose a perfectly matching concept, by iterating the actual process of getting the derivative fixed on a certain plain. We call this concept a partial derivative. Partial derivative is the derivative with respect to one of those variables, while the others are held constant.

 

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Like ordinary derivatives, the partial derivative is defined as a limit. The partial derivative of $n$-variable function $f$ at the point $\left ( a_{1},a_{2},...,a_{n} \right )$ with respect to the $i$-th variable $x_{i}$ is defined as

 $$\frac{\partial f}{\partial x_{i}}\left ( a \right )=\lim_{\Delta x_{i}\rightarrow 0}\frac{f\left ( a_{1},a_{2},...,a_{i}+\Delta x_{i},...,a_{n} \right )-f\left ( a_{1},a_{2},...,a_{n} \right )}{\Delta x_{i}}$$

 

When the limit exists, $f$ is partially differentiable by $x_{i}$ at $\left ( a_{1},a_{2},...,a_{n} \right )$. If $f$ is partially differentiable by all dependent variables at $\left ( a_{1},a_{2},...,a_{n} \right )$, then it is totally differntiable at $\left ( a_{1},a_{2},...,a_{n} \right )$. The partial derivative $\frac{\partial f}{\partial x_{i}}$ can be seen as another function defined on $U$ and can again be partially differentiated. 

2.2.1.3. Partial Derivative

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The partial derivative can be thought of as the rate of change of the function in the $x$-direction. Therefore, partial derivative of $f$ with respect to $x_{i}$ can be seen as the instantaneous rate of change of $f$ with respect to $x_{i}$, setting the change of all other variables as 0. 

 

As mentioned, a partial derivative shares a lot of properties with the derivative of a univariable function.

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A multivariable function can have a higher partial derivative. The $m$-order partial derivative of function $f$ with respect to $x_{1}$, $x_{2}$, ... , $x_{m}$ is defined as $$\frac{\partial }{\partial x_{i1}}\frac{\partial }{\partial x_{i2}}\cdots \frac{\partial }{\partial x_{im}}f=\frac{\partial^m }{\partial x_{i1}\partial x_{i2}\cdots \partial x_{im}}f$$

2.2.1.4. Higher Partial Derivative

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Generally, partial derivation of different variables is not commutative. In other words, $\frac{\partial }{\partial x}(\frac{\partial }{\partial y})f\neq \frac{\partial }{\partial y}(\frac{\partial }{\partial x})f$. However, Alexis Claude Clairaut found an exceptional condition. 

 

Consider a function $f$ defined on $D$ which contains point $(a,b)$. $f_{xy}(a,b)=f_{yx}(a,b)$ if functions $f_{xy}$ and $f_{yx}$ are both differentiable at $D$.

2.2.1.5. Clairaut's Theorem

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