In other words, it depends upon whether you are thinking of "a" as a variable or as a specific value of a variable. If your function has been written as a function of x and then you see f'(a), that is the derivative with respect to the
variable x, evaluated at x= a.
Of course, if you think of a as being the variable, and differentiate f(a) with respect to a, you should get exactly the same thing as if you differentiated f(x) with respect to x, then set x= a.
For example, if the function is
then
and, setting x= a,
. On the other hand, if I replace the variable x with the variable a, I would have
and the derivative with respect to a is
, exactly as before.
Of course, that has to be a letter or something that we
can think of as a variable. If we have
as before and are asked to find
, we would set
and then set
:
.
It would be a terrible
mistake to first set x equal to the
constant and then argue "
is a constant so its derivative is 0". We differentiate with respect to a
variable, not a constant.