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.

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}


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


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

\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}


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} :

\\&=\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

\frac{\part}{\part x}(e^{xy})=ye^{xy}\\\frac{\part}{\part y}(e^{xy})=xe^{xy}

See also

Ad blocker interference detected!

Wikia is a free-to-use site that makes money from advertising. We have a modified experience for viewers using ad blockers

Wikia is not accessible if you’ve made further modifications. Remove the custom ad blocker rule(s) and the page will load as expected.