Average slopeAverage slope
Find the rate of change if it takes 3 hours to drive 210 miles.
What is your average speed or velocity?
( ) (3 0
3
)
0
f f
If it takes 3 hours to drive 210 miles If it takes 3 hours to drive 210 miles
then we averagethen we average
A.A. 1 mile per minute1 mile per minute
B.B. 2 miles per minute2 miles per minute
C.C. 70 miles per hour70 miles per hour
D.D. 55 miles per hour55 miles per hour
Instantaneous slopeInstantaneous slope
What if h went to What if h went to zero?zero?
0'( ) l
( (im
) )h
f x h x
hf x
f
DerivativeDerivative
if the limit exists as one real if the limit exists as one real number. number.
0'( ) l
( (im
) )h
f x h x
hf x
f
DefinitionDefinitionIf f : D -> K is a function then the derivative of f If f : D -> K is a function then the derivative of f
is a new function, is a new function, f ' : D' -> K' as defined above if the limit f ' : D' -> K' as defined above if the limit
exists. exists. Here the limit exists every where except at x = 1Here the limit exists every where except at x = 1
0'( ) l
( (im
) )h
f x h x
hf x
f
Guess at Guess at
0
( ) ( )1lim
1'( )
hf x
f h
h
f
Guess at Guess at
0
( ) ( )1lim
1'( )
hf x
f h
h
f
ThusThus
d.n.e.d.n.e.
0'( ) li
1 1m
( ) ( )h
f fhf
hx
Guess at Guess at
f’(0) – slope of f when x = 0f’(0) – slope of f when x = 0
0'( ) l
( (im
) )h
f x h x
hf x
f
Guess at f ’(3)Guess at f ’(3)
-1.0-1.0
0.490.49
Guess at f ’(-2)Guess at f ’(-2)
-4.0-4.0
2.992.99
Note that the rule is Note that the rule is f '(x) is the slope at the point ( x, f(x) ), f '(x) is the slope at the point ( x, f(x) ), D' is a subset of D, butD' is a subset of D, butK’ has nothing to do with KK’ has nothing to do with K
0'( ) l
( (im
) )h
f x h x
hf x
f
K is the set of distances from homeK is the set of distances from homeK' is the set of speeds K' is the set of speeds K is the set of temperaturesK is the set of temperaturesK' is the set of how fast they rise K' is the set of how fast they rise K is the set of today's profits , K is the set of today's profits , K' tells you how fast they changeK' tells you how fast they changeK is the set of your averages K is the set of your averages K' tells you how fast it is changing. K' tells you how fast it is changing.
0'( ) l
( (im
) )h
f x h x
hf x
f
Theorem If f(x) = c where c Theorem If f(x) = c where c is a real number, then f ' (x) is a real number, then f ' (x) = 0.= 0.
Proof : Lim [f(x+h)-f(x)]/h = Proof : Lim [f(x+h)-f(x)]/h =
Lim (c - c)/h = 0.Lim (c - c)/h = 0.
Examples Examples
If f(x) = 34.25 , then f ’ (x) = 0If f(x) = 34.25 , then f ’ (x) = 0
If f(x) = If f(x) = , then f ’ (x) = 0, then f ’ (x) = 0
If f(x) = 1.3 , find f’(x)If f(x) = 1.3 , find f’(x)
0.00.0
0.10.1
Theorem Theorem If f(x) = x, then f ' (x) = 1. If f(x) = x, then f ' (x) = 1.
Proof : Lim [f(x+h)-f(x)]/h = Proof : Lim [f(x+h)-f(x)]/h =
Lim (x + h - x)/h = Lim h/h = 1Lim (x + h - x)/h = Lim h/h = 1
What is the derivative of x What is the derivative of x grandson?grandson?
One grandpa, one.One grandpa, one.
Theorem If c is a constant,Theorem If c is a constant,(c g) ' (x) = c g ' (x) (c g) ' (x) = c g ' (x)
Proof : Lim [c g(x+h)-c g(x)]/h =Proof : Lim [c g(x+h)-c g(x)]/h =
c Lim [g(x+h) - g(x)]/h = c g ' (x) c Lim [g(x+h) - g(x)]/h = c g ' (x)
Theorem If c is a constant,Theorem If c is a constant,(cf) ' (x) = cf ' (x) (cf) ' (x) = cf ' (x)
( 3 x)’ = 3 (x)’ = 3 or( 3 x)’ = 3 (x)’ = 3 or
If f(x) = 3 x then If f(x) = 3 x then
f ’(x) = 3 times the derivative of xf ’(x) = 3 times the derivative of x
And the derivative of x is . . And the derivative of x is . .
One grandpa, one !!One grandpa, one !!
If f(x) = -2 x then f ’(x) If f(x) = -2 x then f ’(x) = =
-2.0-2.0
0.10.1
TheoremsTheorems
1. (f + g) ' (x) = f ' (x) + g ' (x), and 1. (f + g) ' (x) = f ' (x) + g ' (x), and
2. (f - g) ' (x) = f ' (x) - g ' (x) 2. (f - g) ' (x) = f ' (x) - g ' (x)
1. (f + g) ' (x) = f ' (x) + g ' 1. (f + g) ' (x) = f ' (x) + g ' (x) (x) 2. (f - g) ' (x) = f ' (x) - g ' 2. (f - g) ' (x) = f ' (x) - g ' (x) (x)
If f(x) = 3If f(x) = 322 x + 7, find f ’ x + 7, find f ’ (x)(x)
f ’ (x) = 9 + 0 = 9f ’ (x) = 9 + 0 = 9
If f(x) = x - 7, find f ’ (x)If f(x) = x - 7, find f ’ (x)
f ’ (x) = - 0 = f ’ (x) = - 0 =
55 5
If f(x) = -2 x + 7, find f ’ If f(x) = -2 x + 7, find f ’ (x)(x)
-2.0-2.0
0.10.1
If f(x) = thenIf f(x) = then f’(x) = f’(x) =
Proof : f’(x) = Lim [f(x+h)-f(x)]/h = Proof : f’(x) = Lim [f(x+h)-f(x)]/h =
x1
2 x
If f(x) = then f’(x) = If f(x) = then f’(x) =
A.A. ..
B.B. ..
C.C. ..
D.D. ..
x
0limh
x h x
h
0limx
x h x
h
0limh
x x
h
x h x
h
If f(x) = xIf f(x) = xnn then f ' (x) = n x then f ' (x) = n x (n-(n-
1)1)
If f(x) = xIf f(x) = x44 then f ' (x) = 4 xthen f ' (x) = 4 x33
If If 2
3( )g x
x 23x
2 2 3'( ) (3 ) ' 3( ) ' 3( 2 )g x x x x 3
3
66x
x
If f(x) = xIf f(x) = xnn then f ' (x) = n x then f ' (x) = n xn-1 n-1
If f(x) = xIf f(x) = x44 + 3 x+ 3 x33 - 2 x - 2 x22 - 3 x + 4 - 3 x + 4
f ' (x) = 4 xf ' (x) = 4 x3 3 + . . . .+ . . . .
f ' (x) = 4xf ' (x) = 4x33 + 9 x+ 9 x22 - 4 x – 3 + 0 - 4 x – 3 + 0
f(1) = 1 + 3 – 2 – 3 + 4 = 3f(1) = 1 + 3 – 2 – 3 + 4 = 3
f ’ (1) = 4 + 9 – 4 – 3 = 6f ’ (1) = 4 + 9 – 4 – 3 = 6
3y
If f(x) = xIf f(x) = xnn then f ' (x) = n x then f ' (x) = n x (n-(n-
1)1)If f(x) = If f(x) = xx44 then f ' (x) = 4then f ' (x) = 4 x x33
If f(x) = If f(x) = 44 then f ' (x) = 0then f ' (x) = 0 If If ( ) 3g x x
1
23x1 1 1
2 2 21
'( ) (3 ) ' 3( ) ' 3( )2
g x x x x
1
23 3
2 2x
x
If f(x) = then f ‘(x) =If f(x) = then f ‘(x) =x
1 1
2 21
'( ) ( ) '2
f x x x
1
2 x
Find the equation of the line Find the equation of the line tangent to g when x = 1. tangent to g when x = 1.
If g(x) = xIf g(x) = x33 - 2 x - 2 x22 - 3 x + 4 - 3 x + 4
g ' (x) = 3 xg ' (x) = 3 x22 - 4 x – 3 + 0 - 4 x – 3 + 0
g (1) =g (1) =
g ' (1) =g ' (1) =
If g(x) = xIf g(x) = x33 - 2 x - 2 x22 - 3 x + 4 - 3 x + 4find g (1)find g (1)
0.00.0
0.10.1
If g(x) = xIf g(x) = x33 - 2 x - 2 x22 - 3 x + 4 - 3 x + 4find g’ (1)find g’ (1)
-4.0-4.0
0.10.1
Find the equation of the Find the equation of the line tangent to f when x line tangent to f when x = 1. = 1.
g(1) = 0g(1) = 0
g ' (1) = – 4g ' (1) = – 4
14
0
x
y
4(0 1)xy
( 1)4y x
Find the equation of the line Find the equation of the line tangent to f when x = 1. tangent to f when x = 1.
If f(x) = xIf f(x) = x44 + 3 x+ 3 x33 - 2 x - 2 x22 - 3 x + 4 - 3 x + 4
f ' (x) = 4xf ' (x) = 4x33 + 9 x+ 9 x22 - 4 x – 3 + 0 - 4 x – 3 + 0
f (1) = 1 + 3 – 2 – 3 + 4 = 3f (1) = 1 + 3 – 2 – 3 + 4 = 3
f ' (1) = 4 + 9 – 4 – 3 = 6 f ' (1) = 4 + 9 – 4 – 3 = 6
Find the equation of the Find the equation of the line tangent to f when x line tangent to f when x = 1. = 1.
f(1) = 1 + 3 – 2 – 3 + 4 = 3f(1) = 1 + 3 – 2 – 3 + 4 = 3
f ' (1) = 4 + 9 – 4 – 3 = 6 f ' (1) = 4 + 9 – 4 – 3 = 6
61
3Y
X
Write the equation of the Write the equation of the tangent line to f when x = 0. tangent line to f when x = 0.
If f(x) = xIf f(x) = x44 + 3 x+ 3 x33 - 2 x - 2 x22 - 3 x + 4 - 3 x + 4
f ' (x) = 4xf ' (x) = 4x33 + 9 x+ 9 x22 - 4 x – 3 + 0 - 4 x – 3 + 0
f (0) = write downf (0) = write down
f '(0) = for last questionf '(0) = for last question
Write the equation of the Write the equation of the line tangent to f(x) when x line tangent to f(x) when x = 0.= 0.A.A. y - 4 = -3xy - 4 = -3x
B.B. y - 4 = 3xy - 4 = 3x
C.C. y - 3 = -4xy - 3 = -4x
D.D. y - 4 = -3x + 2y - 4 = -3x + 2
http://www.youtube.com/watch?v=P9dpTTpjymE Derive Derive
http://www.9news.com/video/player.aspx?aid=52138&bw= Kids= Kids
http://math.georgiasouthern.edu/~bmclean/java/p6.html Secant Lines Secant Lines