Is it proper form to write $f'(x)$ in terms of $a$ when using the $x \to a$ method? For first principles derivatives, I solve for $f'(x)$ by doing the following: $$f'(x) = \lim_{x\to a}\frac{f(x)-f(a)}{x-a}$$
Since $x$ approaches $a,$ the final answer will be in terms of $a.$ Is this considered proper form? I had always assumed that if you take the derivative of $f(x),$ you would get an answer in terms of $x.$ Is the $x \to a$ definition of the derivative above (the equation) incorrect? How would I rewrite it so the final answer is in terms of $x,$ if necessary?


Since $x$ approaches $a$, the final answer will be in terms of $a$. Is this considered proper form?

No, this is not proper. The left-hand side is looking for an answer in terms of $x$ while the right-hand side is in terms of $a$. This really doesn't make any sense. In order to fix this, you should make $a$ approach $x$, like so:
$$f'(x)=\lim_{a \to x} \frac{f(x)-f(a)}{x-a}$$
Now, the right-hand side will give you an answer in terms of $x$, so the above is correct.
