(Silly question) Why can't derivative be defined as $[f(a)-f(a+h)]/h$? I'm certain that this is a silly question, but it's one I need to ask anyway, because I'm still learning which parts of calculus are conventions unworthy of serious thought, which parts are trivial to prove, and which parts are worth understanding deeply. The question is:
Why can't the basic derivative (studied in a Calculus I class) be defined as $$\lim_{h\rightarrow 0}\dfrac{f(a)-f(a+h)}{h},$$ equivalently $$\lim_{x\rightarrow a}\dfrac{f(a)-f(x)}{x-a}$$ rather than as $$\lim_{h\rightarrow 0}\dfrac{f(a+h)-f(a)}{h}$$ and $$\lim_{x\rightarrow a}\dfrac{f(x)-f(a)}{x-a}$$ respectively?
I've come up with a potential answer: switching the sign of the numerator (by swapping $$f(x)-f(a)$$ and $$f(a)-f(x)$$ is unacceptable because it gets the sign of the derivative wrong. But then, how can we use knowledge about the sign of the derivative, in order to help us calculate the derivative?
Any help is appreciated.
 A: The derivative is defined this way so that it would give us the slope of the tangent.
Slope is increase in $y$ divided by increase in $x$. The slope of the straight line connecting the two points $(a,f(a))$ and $(x,f(x))$ can be written either as $\frac{f(x)-f(a)}{x-a}$ or as $\frac{f(a)-f(x)}{a-x}$; but it cannot be written as $\frac{f(a)-f(x)}{x-a}$ because now we're calculating decrease in $y$ divided by increase in $x$, which gives the wrong sign.
(Of course, technically speaking there would be nothing wrong in defining $f'(a)=\lim_{x\to a} \frac{f(a)-f(x)}{x-a}$; but then we'd say that the derivative is the negative of the slope of the tangent, and we'd have to reverse all theorems relying on the sign of the derivative.)
A: Why is the derivative defined like this? It is because we are looking for the slope of a function at a point. But this can't be right because when solving for the slope, we need two points, right?
Recall that given two points, $P_1(x_1, y_1)$ and $P_2(x_2,y_2)$, the slope passing through these points are $\frac{y_2 - y_1}{x_2 - x_1}$, or $\frac{y_1 - y_2}{x_1 - x_2}$, both of which are the same. Why is this relevant?
Given a function $f$, let's say we want to find the slope of the line tangent to $f$ at $x = a$. To start, the two points here will be $(a,f(a))$ and the other point will be $(x, f(x))$. This secant will have a slope of $\frac{f(x) - f(a)}{x - a}$. Clearly, we can't let $x = a$. But notice that as $x$ gets closer and closer to $a$,  the secant gets closer and closer to becoming a tangent.
We can write this as $$\lim_{x \to a}\frac{f(x) - f(a)}{x - a}.$$

We go back to the question.

Why can't the basic derivative (studied in a Calculus I class) be defined as $$\lim_{h\rightarrow 0}\dfrac{f(a)-f(a+h)}{h},$$ equivalently $$\lim_{x\rightarrow a}\dfrac{f(a)-f(x)}{x-a}$$ rather than as $$\lim_{h\rightarrow 0}\dfrac{f(a+h)-f(a)}{h}$$ and $$\lim_{x\rightarrow a}\dfrac{f(x)-f(a)}{x-a}$$ respectively?

Because that does not use the formula for the slope correctly. Since the terms in the numerator were swapped and subtraction is not commutative, then the sign will change and hence, give the wrong answer.
