16.4 Finding Derivatives

*We are going to do a little adapting h = x – c and since now On Friday, we had this picture, which lead to cx Q P (c, f (c)) (x, f (x)) x – c  h x  c + h Q (x, f (x))  Q (c + h, f (c + h) and finally we get x – c

If we replace the specific c value with an arbitrary x, we get This has a special name in Calculus! It is called the derivative of f. Notation: f (x) and you read it “f prime of x.” Def: The derivative of a function f is another function f defined by. The domain of f consists of all those values of x in the domain of f for which the limit exists. If the limit exists at x, f is differentiable at x. Finding the derivative is differentiation. (Does it look familiar ?!?)

Ex 1) Find the derivative of f (x) = x 2 – 8x

Once we have found a derivative, we can find the derivative of this new function. The derivative of a derivative is the second derivative and is denoted by f  (x). Ex 2) Find the second derivative of a function in Ex (1).

A pictorial example of how a function & its first & second derivative are related is position, velocity, & acceleration. f time height f time velocity f  time acceleration Algebraic strategy: When using the definition of a derivative to find f you need to eliminate h as a factor of the denominator.

Ex 3) Find the derivative of. Find where f does not have a derivative. No derivative at x = 4. (not defined here)

Ex 3) cont… An important connection is that between differentiability & continuity. - If a function has a derivative at a point, then it is continuous. *The converse is not true! -If a function is continuous, then the function has a derivative. Example: no slope/derivative here!

Homework #1604 Pg 874 #1–21 odd, 23, 26, 29, 31, 33, 35, 36, 40–42