# 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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