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c Roberto Barrera, Fall 2015
Math 142 3.3 The Derivative
The Derivative
The derivative of y = f (x), denoted y0 , f 0 (x),
dy
dx ,
and
d
dx
The derivative of y = f (x) at x = c is defined to be
To find the derivative
1.
2.
3.
Example: Let f (x) =
2
3x+4 .
Find f 0 (x) where x 6= − 34 .
f (x), is defined to be
c Roberto Barrera, Fall 2015
Math 142 2
Example: Below is the graph of a function y = f (x). From this graph, roughly sketch
f 0 (x).
Example: Let f (x) = |x|. Does f 0 (0) exist?
c Roberto Barrera, Fall 2015
Math 142 When does the derivative fail to exist?
• The graph of the function has a corner.
Example: f 0 (0) fails to exist for f (x) = |x|.
• The graph of the function has a vertical tangent.
√
1
Example: f 0 (0) fails to exist for f (x) = 3 x = x 3 .
• The graph of the function has a break (discontinuity).
Derivatives and continuity
If y = f (x) has a derivative at x = c, then f (x) is continuous at x = c.
The converse is not true!
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