Why do we involve the left-hand derivative? I've always thought of slope as "change in y as x increases by 1," so when I think of the slope at a point on a curve, I think of "how would y change as x increases by 1 from this point on." The left-hand derivative kind of seems backwards to me, like "at what rate did y change at in order to get to this point." Why do we force a double-sided limit to exist for differentiability? Why do we need both derivatives to be equal? Please help me understand this.
 A: 
Why do we force a double-sided limit to exist for differentiability?

There's a general pattern in mathematics that the narrower your definitions are, the easier they are to use. If derivatives are defined using two-sided limits, then you can state a theorem like the chain rule as simply "the derivative of the composite is the product of the derivatives". But if derivatives are defined using one-sided limits, then you have to say something like "the derivative of the composite is the product of the derivatives, provided that the inner derivative is nonnegative".
Given this situation, the approach of least resistance is to say that "derivatives" are two-sided limits, and if you want to talk about some other generalization of the derivative concept, you're certainly welcome to do so, but you have to call it something else! And people have done so in various directions: see subderivative, weak derivative, symmetric derivative, Dini derivative, and most importantly, since it's your question: right derivative.
A: Think about the differentiability of the absolute value function. If you want to consider "how fast is $y$ changing as $x$ changes?" at $x=0$, it depends on which "side" of the absolute value function you look at. For negative $x$, $y$ is decreasing as $x$ increases, but for positive $x$, $y$ increases as $x$ increases. So at $x=0$ it doesn't really make sense to consider the rate at which $y$ changes as $x$ changes, as it depends on which "side" the change in $x$ occurs. Hopefully this helps you to see why we require both the left- and right-hand limits of the difference quotient to exist and be equal.
