This is from my exam:
1) Calculate partial derivative $f'(10)$ of the function: $$f(x)=\frac{1-\log x}{1+\log x}.$$
This is a function of only one variable, why do they use the term 'partial' ?
Are the terms derivative and differential interchangeable for one variable functions.
Is it correct to write (for this example):
$f'(10)=\frac{\mathrm{d}f}{\mathrm{d}x}(10)$ or $f'(10)=\frac{\partial f}{\partial x}(10)$
Best Answer
As pointed out correctly above, if $f$ only depends on one variable, you can write it either way, though the expression $\frac{df}{dx}$ is found more often with functions of only one variable. Many use it as a way of telling apart a derivative of a single variable function $f$ ($\frac{df}{dx}$) and a partial derivative of a function $f$ with respect to the variable $x$ ($\frac{\partial f}{\partial x}$), implying that there are other variables in $f$ as well.
They might have used the expression "partial" to try to mess with your head, but I don't really see why anyone would actually call it like this elsewise. It may also be possible that the function was supposed to contain multiple variables and that they've written it wrong, but then again, they did not specify which variable, leading me to believe that it was sort of a trick question and not a typo.
The expression "differential" is used to denote a infinitely small (infinitesimal) change in a quantity. Of course you do that when taking a derivative, but that doesn't mean that you can use these two words interchangeably at all times. Check out the wiki links, they might be of some help:
http://en.wikipedia.org/wiki/Derivative
http://en.wikipedia.org/wiki/Differential_(infinitesimal)
Sincerely,
SDV