1. 2.2 The derivative as a function
In Section 2.1 we considered the derivative at a fixed number a.
Now let number a vary. If we replace number a by a variable x, then the derivative can be interpreted as a function of x :
Alternative notations for the derivative:
D and d / dx are called differentiation operators.
dy / dx should not be regarded as a ratio.

2. The derivative is the slope of the original function.

4. Differentiable functions
A function f is differentiable at a if f ′(a) exists. It is differentiable on an open interval (a,b) [ or (a,) or (- , a) or (- , ) ] if it is differentiable at every number in the interval.
Theorem: If f is differentiable at a, then f is continuous at a.
Note: The converse is false: there are functions that are continuous but not differentiable.
Example: f(x) = | x |

5. To be differentiable, a function must be continuous and smooth.
Derivatives will fail to exist at:
corner
cusp
discontinuity
vertical tangent

6. Higher Order Derivatives:
is the first derivative of y with respect to x.
is the second derivative.
(y double prime)
is the third derivative.
We will learn later what these higher order derivatives are used for.
is the fourth derivative.