1. 2.1 The derivative and the tangent line problem

2. Definition of the derivative of a function
The derivative of f at x is given by f’(x) = lim f(x + ∆x) – f(x) ∆x -> 0 ∆x provided the limit exists. For all x for which this limit exists, f’ is a function of x.

3. Notation for derivatives:
f’(x) “f prime of x” dy “the derivative of y with respect to x” dx “ dy – dx” y’ “y prime” d [f(x)] dx Dx[y]

4. dy = lim ∆y
dx ∆x ∆x
= lim f(x + ∆x) – f(x)
∆x ∆x
= f’(x)

5. Alternative Form of a Derivative
f’(c) = lim f(x) - f(c)
x c x - c

6. f’(c) = lim f(x) - f(c) x c- x - c f’(c) = lim f(x) - f(c) x c+ x - c
The existence of the limit requires that the one-sided limits exist and are equal.
f’(c) = lim f(x) - f(c)
x c x - c
(The derivative from the left)
f’(c) = lim f(x) - f(c)
x c x - c
(The derivative from the right)

7. f is differentiable on the closed interval [a, b] if it is differentiable on (a, b) and if the derivative from the right at a and the derivative from the left at b both exist.

8. Example:
f(x) = |x – 2|

9. Conditions where a function is not differentiable:
1. At a point at which a graph has a sharp turn.
2. At a point at which a graph has a vertical tangent line.
3. At a point at which the function is not continuous.

10. Theorem: Differentiability Implies Continuity
If “f” is differentiable at x = c, then f is continuous at x = c.