Apollonius of Perga
262 – 190 B.C.
Apollonius was a Greek mathematician known as 'The Great
Geometer'. His works had a very great influence on the development
of mathematics and his famous book Conics introduced the terms
parabola, ellipse and hyperbola.
Secant vs Tangent Lines Definition 1
y
Slope of Secant Line
msecant
Secant Line
y2  y1

x2  x1
Average Rate of Change
(a, f (a))
Tangent Line
msecant
f ( x)  f (a)

xa
Difference Quotient
f (a)
(x, f (x))
msecant
y = f (x)
f ( x)  f (a)

xa
Instantaneous Rate of Change
a
x
a, x
x
mtangent
f ( x)  f (a)
 lim
xa
xa
1
Example 1: Using the definition above, find the slope of the tangent line for f ( x)  x 2  5 x at a  3.
x2  5x  6
 lim
x 3
x 3
f ( x)  f (3)
m  lim
x 3
x 3
f (3)
f ( x)
 lim
x 3
x 2  5 x   32  5  3 
x 3
( x  3)( x  2)
x 3
x 3
 lim
1
Secant vs Tangent Lines Definition 2
y
Slope of Secant Line
msecant
Secant Line
y2  y1

x2  x1
Average Rate of Change
(a, f (a))
Tangent Line
msecant
f ( a  h)  f ( a )

aha
Difference Quotient
f (a)
(a + h, f (a + h))
msecant 
y = f (x)
h
a
a+h
a, a  h
f ( a  h)  f ( a )
h
Instantaneous Rate of Change
x
mtangent
f ( a  h)  f ( a )
 lim
h 0
h
2
Example 2: Using the definition above, find the slope of the tangent line for f ( x)  x 2  5 x at a  3.
m  lim
h 0
f (3  h)  f (3)
h
f (3  h)
 lim
h 0
f (3)
(3  h)2  5(3  h)   32  5  3 
h
9  6h  h 2  15  5h  6
 lim
h 0
h
h  h2
 lim
h 0
h
 lim(1  h)
h 0
1
Find the equation of the tangent line at a = 3.
1) Slope
m |a 3  1
2) Point of Tangency
(3, f (3))  (3, 6)
y  x9
3) Formula
y  y1  m( x  x1 )
y  (6)  1( x  3)
y  x9
(3, 6)
Secant vs Tangent Lines Definitions 3 and 4
Example 3: Find the derivative of f ( x)  x .
f ( x)  lim
h 0
f ( x  h)  f ( x )
h
xh  x xh  x

 lim
h 0
xh  x
h
xhx
h 0 h( x  h 
x)
 lim
h
h 0 h( x  h 
x)
 lim

1
x0  x
Hence,
f ( x) 
1
2 x
Find the equation of the tangent line at a = 4.
1) Slope
f ( x) 
1
2 x
, so m  f (a)  f (4) 
1
2 4

1
4
2) Point of Tangency (a, f (a)) (POT)
(4, f (4))  (4, 2)  f ( x)  x
y
3) Formula
y  y1  m( x  x1 )
1
y  2  ( x  4)
4
1
y  x 1
4
(4, 2)
1
x 1
4
“The derivative of f with respect to x is …”
There are other forms that represent the derivative of the function f.
f ( a  h)  f (a )
f '(a)  lim
h 0
h
f ( x)  f (a)
f '(a)  lim
x a
xa
y
f '( x)  lim
x  0 x
f ( x2 )  f ( x1 )
f '( x1 )  lim
x2  x1
x2  x1
There are many ways to write the derivative of
y  f  x
f  x
y
“f prime x” or “the derivative of f with respect to x”
“y prime”
dy
dx
“dee why dee ecks”
or
“the derivative of y with
respect to x”
df
dx
“dee eff dee ecks”
or
“the derivative of f with
respect to x”
d
f  x  “dee dee ecks uv eff uv ecks”
dx
( d dx of f of x )
or “the derivative
of f of x”

dx does not mean d times x
dy does not mean d times y
dy
does not mean dy  dx
dx
(except when it is convenient to think of it as division.)
df
does not mean df  dx
dx
(except when it is convenient to think of it as division.)
d
d
f  x  does not mean
times f  x 
dx
dx
(except when it is convenient to treat it that way.)
Example 4
a.)
(1  h)3  1
b.) lim
h 0
h
1
Example 5: Consider the function f ( x) 
.
2x  5
a) Find f ( x).
9
b) Find the slope of the tangent line to the curve when a  .
2
c) Find the equation of the tangent and normal lines to the
9
curve when a  .
2