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The partial derivative extends the concept of the derivative in the one-dimensional case by studying real-valued functions defined on subsets of $ \R^n $ . Informally, the partial derivative of a scalar field may be thought of as the derivative of said function with respect to a single variable.

Definition
Let $ f:D\to\R $ , $ D\subset\R^n $ , be a function. Let $ f(x_1,\ldots,x_n) $ denote the function value of $ f $ at $ (x_1,\ldots,x_n)\in D $ . We define the partial derivative of $ f $ with respect to $ x_i $ ($ i\in\{1,\ldots,n\} $) to be the number
$ \frac{\part f(x_1,\ldots,x_n)}{\part x_i}=\lim_{h\to0}\frac{f(x_1,\ldots,x_i+h,\ldots,x_n)-f(x_1,\ldots,x_n)}{h} $

whenever the limit exists and is finite.

We thus associate with $ f $ a function $ \frac{\part f}{\part x_i}:D^{(i)}\to\R $ , called the partial derivative of $ f $ with respect to the $ i $-th variable, where $ D^{(i)} $ is the subset of $ D $ where the limit above exists.

In $ \R^2 $ it is common to write $ (x,y) $ in place of $ (x_1,x_2) $ , and we usually speak of the partial derivative of $ f $ with respect to $ x $ or $ y $ , defined by

$ \begin{align}\frac{\part f(x,y)}{\part x}&=\lim_{h\to0}\frac{f(x+h,y)-f(x,y)}{h} \\\frac{\part f(x,y)}{\part y}&=\lim_{h\to0}\frac{f(x,y+h)-f(x,y)}{h}\end{align} $

respectively.

Example: dot product with fixed vector

Fix a vector $ (\alpha_1,\ldots,\alpha_n)\in\R^n $ , and define a function $ f:\R^n\to\R $ by

$ f(x_1,\ldots,x_n=(\alpha_1,\ldots,\alpha_n)\cdot(x_1,\ldots,x_n)=\sum_{i=1}^n\alpha_ix_i $

Then the partial derivative of $ f $ with respect to $ x_i $ is equal to $ \alpha_i $ :

$ \begin{align} \frac{\part f(x_1,\ldots,x_n)}{\part x_i}&=\lim_{h\to0}\frac{f(x_1,\ldots,x_i+h,\ldots,x_n)-f(x_1,\ldots,x_n)}{h} \\&=\lim_{h\to0}\frac{\alpha_1x_1+\cdots+\alpha_i(x_i+h)+\cdots+\alpha_nx_n-(\alpha_1x_1+\cdots+\alpha_nx_n)}{h} \\&=\lim_{h\to0}\frac{\alpha_i(x_i+h)-\alpha_ix_i}{h}=\lim_{h\to0}\alpha_i=\alpha_i \end{align} $

Computation

We have shown that

$ \frac{\part}{\part x_i}(\alpha_1x_1+\cdots+\alpha_ix_i+\cdots+\alpha_nx_n)=\alpha_i $

This is an example of a property that can be shown to hold in general: when taking the partial derivative of a function with respect to some variable, one can differentiate as though all the other variables were constants in an ordinary derivative. That is, if we wish to compute the partial derivative of a function $ f:\R^n\to\R $ at a point $ (x_1,\ldots,x_n)\in\R^n $ with respect to $ x_i $ , we may introduce another function $ g:\R\to\R $ given by $ g(x)=f(x_1,\ldots,x,\ldots,x_n) $ , where the $ x $ is in the $ i $-th place, and all the other components are held fixed. It is a trivial matter to verify that $ g'(x)=\frac{\part f(x_1,\ldots,x_n)}{\part x_i} $ :

$ \begin{align}g'(x) &=\lim_{h\to0}\frac{g(x+h)-g(x)}{h} \\&=\lim_{h\to0}\frac{f(x_1,\ldots,x+h,\ldots,x_n)-f(x_1,\ldots,x,\ldots,x_n)}{h} \\&=\frac{\part f(x_1,\ldots,x_n)}{\part x_i}\end{align} $

Example: two variables

Recall from the one-dimensional theory that if $ \alpha $ is any constant, then $ \frac{d}{dx}(e^{\alpha x})=\alpha e^{\alpha x} $ .

Now define a function $ f:\R^2\to\R $ by $ f(x,y)=e^{xy} $ . The discussion above allows us to use the one-dimensional theory to compute $ \frac{\part f}{\part x} $ and $ \frac{\part f}{\part y} $ with ease: we simply note that in each of these derivatives, we may treat the second variable as a constant, and evaluate the derivative like in the one-dimensional case. Hence

$ \begin{align} \frac{\part}{\part x}(e^{xy})=ye^{xy}\\\frac{\part}{\part y}(e^{xy})=xe^{xy} \end{align} $

See also

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