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Lecture Notes for Section The Derivative Calculus

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					Calculus 1 Lecture Notes                 Section 2.2                             Page 1 of 5

Section 2.2: The Derivative

Big Idea: The slope of the tangent line to a function at a point is called the derivative of the function at
that point.

Big Skill: You should be able to compute the derivative of a function at a point or the derivative
function in general by evaluating the difference quotient limit (i.e., the limit of the secant line slopes).

Definition 2.1: The Derivative at a Point
The derivative of the function f(x) at x = a is defined as
                           f a  h  f a
          f '  a   lim                    ,
                      h 0         h
provided the limit exists.
If the limit exists, we say that f is differentiable at x = a.

Alternative Definition of the Derivative at a Point
                 f b  f  a 
f '  a   lim
            b a     ba

Practice:
   1. Find f (a) for f(x) = 4x – 9 at a = 3 using both definitions of the derivative.




    2. Find the derivative of f(x) = 3x2 – 2x + 1 at x = -2 using both definitions of the derivative.
Calculus 1 Lecture Notes               Section 2.2                            Page 2 of 5

Definition 2.2: The Derivative Function
The derivative of the function f(x) is defined as
                         f  x  h  f  x
        f '  x   lim                      ,
                    h 0          h
provided the limit exists.
The process of computing a derivative is called differentiation.
Notice that differentiation produces a function whose output is the derivative of the function f(x) at
any value of x.

Practice:
   3. Find f (x) for f(x) = 4x – 9.




   4. Find f (x) for f(x) = 3x2 – 2x + 1.




   5. Find f (x) for f(x) = 2x3 + 4x2 + x + 1.
Calculus 1 Lecture Notes                   Section 2.2   Page 3 of 5

                                 1
  6. Find f (x) for f  x       .
                                 x




                                  3
  7. Find f (x) for f  x         .
                                 x2




  8. Find f (x) for f  x   x .




  9. Find f (x) for f  x   3 x  1 .
Calculus 1 Lecture Notes                 Section 2.2                  Page 4 of 5

Skill: Sketching graphs of f(x) or f (x) given the other function.
Things to notice:
    Extrema of a graph have a slope / derivative of zero.
    Increasing regions of the graph have a positive derivative.
    Decreasing regions of the graph have a negative derivative.

Practice:
   10. Sketch the derivative of the function graphed below.




    11. Sketch the function whose derivative is graphed below.




Alternative derivative notations:
If y = f(x), then we also write the derivative of f(x) as:
                  dy df   d
 f   x   y           f  x
                  dx dx dx
Calculus 1 Lecture Notes                          Section 2.2                   Page 5 of 5

Theorem 2.1: Relationship Between Differentiability and Continuity.
If f(x) is differentiable at x = a, then f(x) is continuous at x = a.
Proof:
Continuity at x = a requires lim f  x   f  a  . This is what we have to show, starting from the given
                                         x a
that f(x) is differentiable 
                                          f  x  f a          
        lim  f  x   f  a    lim                   x  a
        x a                        x a
                                               xa                
                                          f  x  f a 
                                    lim                  lim  x  a 
                                     x a      xa        x a
                                   f 'a  0
        lim  f  x   f  a    0
        x a

 lim  f  x    lim  f  a    0
 x a             x a

                  lim  f  x    f  a 
                  x a
QED.

This theorem captures the idea that for a derivative to exist, both one-sided limits used in the derivative
must exist. Examples where the one-sided derivatives are different are functions that have sharp
corners, jump discontinuities, vertical asymptotes, or cusps. The derivative also doesn’t exist where a
function has a vertical tangent line, even though the function is still continuous at that point.

Practice:
Find the equation of the tangent line to the following curves at the point specified.:
                                 2 x if x  0
   12. Find f (0) for f  x                .
                                 3x if x  0




                                           1
    13. Find f (0) for f  x               .
                                           x2

				
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