Guys I am sorry if this question has been asked before, but it has been pestering me for long time. It is about the notation of derivatives using 'prime'.
If f(x) is a function of x, then does f'(a) represent f(a) differentiated with respect to a, or does it represent f(x) differentiated with respect to x where x=a or are they both the same thing?
Forgive the lack of Latex sophistication.
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.