What is the difference between \dfrac and \frac?

Mainly it's a difference of size:

• \dfrac means that the fraction is set in displaystyle

• \tfrac means that the fraction is set in textstyle

• with \frac: the actual context implies the decision above

For example, if you have a complicated mathematical expression in the middle of some array environment (which, by default, employs text-style math), you can use \dfrac to force the use of displaystyle math mode for the fraction

\dfrac and \tfrac are from the amsmath package

Aside: TeX provides four math "styles": display style, text style, script style, and scriptscript style. scriptstyle is generally meant for level-1 subscripts and superscripts, and scriptscriptstyle is for level-2 subscripts and superscripts.

\dfrac forces the fraction into display mode, no matter which mode it is in already. On the other hand \tfrac forces it to be in text mode.

Usually \tfrac is used much more than \dfrac. Of course \frac should be used in almost every case but \tfrac is handy for coefficients, as shown by the code below.

\[
   f(x) =  \frac{1}{2} x^2 = \dfrac{1}{2} x^2 = \tfrac{1}{2} x^2
\]

$
   f(x) =  \frac{1}{2} x^2 = \dfrac{1}{2} x^2 = \tfrac{1}{2} x^2
$