I found out the answer to why there are two very similar definitions of the derivative. One has *x-a* on the bottom, with a limit *x → a*. The other one has *h* on the bottom, with a limit *h → 0*. So these two things are the same, just different ways of thinking about the same concept. *h* represents the difference between x and a, so the smaller the difference gets the closer h is to 0. The official definition is the one with h.

A function F is differentiable on an interval if F is differentiable for every point a.

**Key Properties of Differentiability**

1. F is differentiable if F is continuous

2. F is differentiable if F is smooth.

For |x| write it in bracket form!

Q1: Is f(x) continuous? Yes because a limit exists for all point a.

Q2: Is f(x) differentiable? Use the definition of derivative (h → 0) and see if the same from left and right. This derivative from the left and the right is not the same, it is -1 and 1. F is not differentiable at 0.

**Rule**

Tan, always break apart.

**Property – Write this on a card**

1. If f’(x) < 0 is decreasing near x and inverse: a. f’’(x) > 0 f is concave up.

b. f’’(x) < 0 f is concave down.