By: Jane Kim , Period 4, 2007

Source: Mr. Wiencek’s Noted from Class

The Definition of Derivative

f(x)

(x,0)

(x,f(x))

((x+h),f(x+h))

h

((x+h),0)

Limit Definition of a Derivative

h 0limf ‘(x) =

f(x+h) – f(x)

h

f ‘(x) = limh 0

f(x+h) – f(x)

h

Remember!

• Don’t forget to write out the Limit Definition of Derivative

• Remember to write everytime limh 0

Example

f(x) = 5x + 3

F(x+h) = 5(x+h) + 3

= 5x + 5h + 3

5x + 5h + 3 – (5x + 3)

h

Example continued

5x + 5h + 3 – 5x – 3

h

5h

h

= 5Answer: 5

Power Rule, Slopes of Tangent Lines

• f ‘(x) F Prime of x

• y’ y prime

• dy

dx

• d

dx

dy dx

Derivatives with respect to x

Common Powers

x = x

5

x = x

1x = x

1

x3= x

LN Yx = x

x = x43

x

x

x

x

x

x

1/2

1/5

-1

- 3

4

4/3

Power Rule

y = x

y’ = x*– 1

Example

y = 3x - x + 2

y’ = 2(3)x – 1(-1) + 0(2)

= 6x + (-1)x + 0x

= 6x – 1

2

2 - 1 1 - 1 0 - 1

0 -1

Answer: = 6x – 1

dy

dxdy

dx

Remember!

• Derivatives = Slope

Example

y = 2x x = 0,1,3,-4

f(0) = 4(0) = 0

f(1) = 4(1) = 4

f(3) = 4(3) = 12

f(-4) = 4(-4) = -16

dy

dx = 4x

2

Graphs & Using the Derivative to find Slope

Tangent Line

Slope = m

Normal Line

Slope = 1m

Example

y = 2x + 3Find the equation ofa) The tangent at 1b) The normal at 1

y = 2x + 3y’= 6x + 3y(1) = 6(1) + 3

= 6 + 3 = 9

3

3

2

(1, ?)

**Derivatives = Slope Slope = 9

Example continued

To find y: plug x = 1 back into the original equation, y = 2x + 3

y = 2(1) + 3(1)

= 2 + 3

= 5

so (1,5)

3

3

(1, ?)

Example continued

Tangent equation:

y – y = m(x – x )

y – 5 = 9(x – 1)

y – 5 = 9x – 9

y = 9x – 4

Normal equation:

y – y = - 1/9(x – x )

y – 5 = - 1/9(x – 1)

y – 5 = -1/9x + 1/9

y = -1/9x + 46/9

1 1 1 1

THE END