3.2 The Derivative as a Function

This section introduces the derivative as a function - previously only considered the derivative at a particular point, a.

The Derivative of f(x)

f '(x) =

Note:  The domain of f '(x)  <  The domain of f(x)

f '(x) is a new function

Finding derivatives visually

#2  Use the graph to estimate the value of each derivative.  The sketch the graph of f' .

Trace or copy the graph of the given function f.  (Assume that the axes have equal scales)  Then use the Method of example 1 to sketch the graph of f ' below it.

Match the graph of each function in a-d, with the graph of its derivative in I-IV.

I.                                  a)

II.                                  b)

III.                                    c)

IV.                           d)

Definition

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.

NOTATION

Examples:

Find the derivative of the function using the definition of derivative.  State the domain of the function and the domain of its derivative.

, find f '(x).    Then look at both graphs.

#22

#24

Theorem

If f is differentiable at a, then f is continuous at a.

***CONVERSE IS NOT NECESSARILY TRUE***

Just because a function is continuous at a does NOT mean that the function is differentiable at a!!

Classic example:  f(x) = |x|

When is a function NOT differentiable    (more graphs pg. 129)

When the limit does not exist

Sharp Corners              Discontinuities              Vertical Tangents

Example:

Make a careful sketch of the graph of the sine function and below it sketch the graph of its derivative in the same manner as earlier.  Can you guess what the derivative of the sine function is from its graph?

Second Derivative - is the derivative of the first derivative

f ''  is the rate of change in the "rate of change"

For s(t), a distance function, the 2nd derivative is acceleration

s(t);    v(t)=s'(t);  a(t) = v'(t) = s''(t)

Examples

For the function  , we found the first derivative above.

Now find the second derivative.

The figure below shows the graphs of f, f ‘, and f ‘’. Identify each curve.

Assignment:  3.2 page 131; 1-9 odd, 17-27 odd, 33, 35, 39, 43