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.

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]

dy = lim ∆ydx ∆x 0 ∆x

= lim f(x + ∆x) – f(x) ∆x 0 ∆x

= f’(x)

Alternative Form of a Derivative

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

f’(c) = lim f(x) - f(c) x c+ x - c

(The derivative from the left)

(The derivative from the right)

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.

Example:

f(x) = |x – 2|

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.

Theorem: Differentiability

Implies Continuity

If “f” is differentiable at x = c, then f is continuous at x = c.