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.