MATH 131-503 Fall 2015 2.7
The derivative of f is defined as f
0
( x ) = lim h → 0 f ( x + h ) − f ( x ) h if the limit exists.
Examples:
1. (p. 156) (a) Find the derivative of f ( x ) =
√
1 + 2 x using the definition of derivative.
c Wen Liu
(b) State the domain of f and f
0
.
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MATH 131-503 Fall 2015 2.7
2. The graph of a function f is given below. Find the graph of f
0
.
c Wen Liu
3. The graph of a function f is given below. Find the graph of f
0
.
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MATH 131-503 Fall 2015 2.7
c Wen Liu
4. (p. 155) Match the graph of each function in the second row with the graph of its derivative in the first row. Give reasons for your choices.
(a) (b) (c) (d)
Other notations: If we use the traditional notation y = f ( x ) to indicate that the independent variable is x and the dependent variable is y , then some common alternative notations for the derivative are as follows: f
0
( x ) = y
0 dy
= dx df
= dx
= d dx f ( x ) = Df ( x ) = D x f ( x )
The symbols D and d dx are called differentiation operators because they indicate the operation of differentiation , which is the process of calculating a derivative.
If we want to indicate the value of a derivative dy dx at a specific number a , we use the notation f
0
( a ) = dy dx x = a or f
0
( a ) = dy dx x = a
A function f is differentiable at a if f
0
( 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.
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MATH 131-503 Fall 2015 2.7
How Can a Function Fail to be Differentiable?
(p. 152) c Wen Liu
The last two graphs show that the curve has a vertical tangent line when x = a ; that is, f is continuous at a and lim x → a
| f
0
( x ) | = ∞ .
Theorem: If f is differentiable at a , then f is continuous at a .
Note: The converse of the theorem is false; that is, there are functions that are continuous but not differentiable. Can you give an example?
Higher Derivatives
If f is a differentiable function, then its derivative f
0 its own, denoted by ( f
0
)
0
= f
00
. This new function f
00 is also a function, so f
0 is called the may have a derivative of second derivative of f because it is the derivative of the derivative of f . Using Leibniz notation, d dx dy dx d
2 y
= dx 2 which can be interpreted as the slope of the curve y = f
0
( x ) at the point ( x, f
0
( x )). In other words, it is the rate of change of the slope of the original curve y = f ( x ).
In general, the n th derivative of f is denoted by f ( n ) times. If y = f ( x ), we write and is obtained from f by differentiating n y
( n )
= f
( n )
( x ) = d n y dx n
Applications: If s = s ( t ) is the position function of an object that moves in a straight line, its first derivative represents the velocity v ( t ) of the object v ( t ) = s
0
( t ) = ds dt
The instantaneous rate of change of velocity with respect to time is called the acceleration a ( t ) of
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MATH 131-503 Fall 2015 2.7
the object a ( t ) = v
0
( t ) = s
00
( t ) = d 2 s dt 2
Examples:
5. If s ( t ) = t 3 − t , find the acceleration after 2 seconds.
c Wen Liu
6. (p. 157) The figure shows the graphs of four functions. One is the position function of a car, one is the velocity of the car, one is its acceleration, and one is its jerk. Identify each curve and explain your choices.
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