Partial Derivatives
Why
We want to talk about how a function of multiple real-valued arguments changes with respect to changes in its arguments.1
Definition
Let $f: \R ^d \to \R $
For $i = 1,\dots ,d$, the
$i$th partial derivative
of $f$ is the function $g_i: \R ^d \to \R $
defined by
\[
g_i(x) = \lim_{h \to 0} \frac{f(x + he_i) - f(x)}{h}
\]
Gradient
The gradient of a multivariate function is the vector-valued function whose $i$th component is the the partial derivative of the function with respect to its $i$th argument.
Notation
Let $f: \R ^n \to \R $. The gradient of $f$ is frequently denoted $\nabla f$. It is understood that $(\nabla f) \in \R ^d \to \R ^d$. An alternative notation is to use that similar for single derivatives and to denote the gradient (sometimes called derivative) of $f$ by $f'$ (assuming it exists). It is important to here note that although when $g: \R \to \R $, $g' \in (\R \to \R )$, (and so is another function from and to reals) when $f: \R ^d \to \R $, $f' \in \R ^d \to \R ^d$, and so is a vector-valued (not a real-valued) function.
There is (unfortunately) much notation for the individual partial derivatives; most of which we shall not (fortunately) have occasion to use in these sheets. One popular usage is the use of the $\partial $ symbol, read aloud as “partial.” For example, if $f: \R ^2 \to \R $ is a function of two arguments, each being referred to as $x$ and $y$, then $\partial _x f$ denotes the partial derivative of $f$ with respect to $x$ and $\partial _y f$ denotes the partial derivative of $f$ with respect to $y$. It is understood that $(\partial _x f) \in \R ^d \to \R $. and likewise for $\partial _y f$. Another popular usage is $\partial f/\partial x$ for $\partial _x f$ and $\partial f/\partial y$ for $\partial _y f$. We will almost exclusively prefer the gradient notation.
- Future editions will modify this sheet. ↩︎