Chapter 17.2 The Derivative

How do we use the derivative??

When graphing the derivative, you are graphing the slope of the original function.

Graphing

Which is the f(x) and which is f’(x)?The derivative is 0 (crosses the x-axis)

wherever there is a horizontal tangentY1 = f(x)Y2 = f’(x)

Notation

There are lots of ways to denote the derivative of a function y = f(x).

f’(x) the derivative of f the derivative of f with y’ y prime respect to x.

the derivative of y the derivative of f at x with respect to x. dxdy

dxdf

)(xfdxd

dx does not mean d times x !

dy does not mean d times y !

dydx does not mean !dy dx

(except when it is convenient to think of it as division.)

dfdx

does not mean !df dx

(except when it is convenient to think of it as division.)

(except when it is convenient to treat it that way.)

d f xdx

does not mean times !ddx

f x

Constant Rule

If f(x) = 4 If f (x) = π

If the derivative of a function is its slope, then for a constant function, the derivative must be zero.

examples: 3y 0y then f ’(x) = 0 then f ’(x) = 0

Power Rule

examples: 4f x x

34f x x

8y x

78y x

Power Rule Examples

Example 1: Given f(x) = 3x2, find f’(x)

Example 2: Find the first derivative given f(x) = 8x

Example 3: Find the first derivative given f(x) = x6

Example 4: Given f(x) = 5x, find f’(x)

Example 5: Given f(x) = , find f’(x)

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1x 3

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f '(x) =−3x 4

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f '(x) = 6x

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f '(x) = 8

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f '(x) = 6x 5

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f '(x) = 5

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f (x) = x −3

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f '(x) = −3x−4

Sum or Difference Rule

(Each term is treated separately)

4 12y x x 34 12y x

4 22 2y x x

34 4dy x xdx

EXAMPLES:

Sum/Difference Examples

EX 1: Find f’(x), given:

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f (x) = 5x 4 − 2x 3 − 5x 2 + 8x +11

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f '(x) = 20x 3 − 6x 2 −10x + 8

Sum/Difference Examples

Find p’(t) given

Rewrite p(t): 1

4 12p(t) 12t 6t 5t

13 22

32

p'(t) 48t 3t 5t3 5p'(t) 48t t t

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p(t) =12t 4 − 6 t +5t

Product Rule

2 33 2 5d x x xdx

5 3 32 5 6 15d x x x xdx

5 32 11 15d x x xdx

4 210 33 15x x

2 3x 26 5x 32 5x x 2x

4 2 2 4 26 5 18 15 4 10x x x x x 4 210 33 15x x

One example done two different ways:

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dydx

= f '(x)g(x) + g'(x) f (x)

Product Rule - Example

Let f(x) = (2x + 3)(3x2). Find f’(x)

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f '(x) = (2x + 3)(6x) + (3x 2)(2)

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f '(x) =12x 2 +18x + 6x 2

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f '(x) =18x 2 +18x

Product Rule

Find f’(x) given that

2f (x) x 3 x 5x

1 1

22 21x 3 2x 5 x 5x x2

3 12 25 15x 6x x 152 2

Chain Rule Outside/Inside method of chain rule

insideoutside derivative of outside wrt inside

derivative of inside

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dydx

=ddx

f g(x)( ) = f ' g(x)( ) • g'(x)

Outside/Inside method of chain rule example

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ddx

3x 2 − x +1( )1

3 ⎛ ⎝ ⎜ ⎞

⎠ ⎟= f ' g(x)( ) • g'(x)

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13

3x 2 − x +1( )−2

3 • (6x −1)

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2x − 13

3x 2 − x +1( )2

3

More examples together:2 71) ( ) (3 5 )f x x x 2 232) ( ) ( 1)f x x

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f '(x) = 7 3x − 5x 2( )6(3 −10x)

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f (x) = x 2−1( )23

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f '(x) =23

x 2 −1( )−1

3(2x)

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f '(x) =

43

x

x 2 −1( )1

3

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3) f (t) =−7

2t − 3( )2€

f '(x) = 3x − 5x 2( )6(21 − 70x)

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f (x) = −7 2t − 3( )−2

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f '(x) =14 2t − 3( )−3(2)

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f '(x) =28

2t − 3( )3

Quotient Rule

3

2

2 53

d x xdx x

2 2 3

22

3 6 5 2 5 2

3

x x x x x

x

EXAMPLE:

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f '(x) =2x 4 + 23x 2 +15 −10x

x 4 + 6x 2 + 9

Quotient Rule Example

Find f’(x) if 2x 1f(x) 4x 3

2

4x 3 (2) 2x 1 44x 3

210

4x 3

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10

16x 2 + 24 x + 9

1

2 2 2( ) 1f x x x 2 2( ) 1f x x x

1 1

2 2 2 22 2( ) 1 1d df x x x x xdx dx

1 1

2 2 22 21( ) 1 ( 2 ) 1 22

f x x x x x x

132 2

12 2

( ) 1 2

1

xf x x x

x

1 12 22 23 3 2 3 3

1 1 1 12 2 2 22 2 2 2

1 2 1 (1 )2 2 2( )

1 1 1 1

x x xx x x x x x xf x

x x x x

3

12 2

3 2( )

1

x xf x

x

3

23 1( )

3xf x

x

2

2 23 1 3 1( ) 3

3 3x d xf x

dxx x

2 2

2 22

3 1 ( 3)3 (3 1)(2 )33 3

x x x xx x

2 2 2

2 22

3 1 (3 9) (6 2 )33 3

x x x xx x

2 2 2

2 22

2 2

42

3(3 1) ( 3 2 9)3 1 3 9 6 233 3 3

x x x xx x

x x x

x

Quotient rule

Product & Quotient Rules

Find

2

27x 9 3 4x (5) (5x 1)( 4) (3 11x 20x )(7)

(7x 9)

x

3 4x 5x 1D 7x 9

x x2

7x 9 D 3 4x 5x 1 3 4x 5x 1 D 7x 9(7x 9)

22

140x 360x 120(7x 9)

Applications

Marginal variables can be cost, revenue, and/or profit. Marginal refers to rates of change.

Since the derivative gives the rate of change of a function, we find the derivative.

Application Example

The total cost in hundreds of dollars to produce x thousand barrels of a beverage is given by

C(x) = 4x2 + 100x + 500Find the marginal cost for x = 5

C’(x) = 8x + 100; C’(5) = 140

Example Continued

After 5,000 barrels have been produced, the cost to produce 1,000 more barrels will be approximately $14,000

The actual cost will be C(6) – C(5): 144 or $14,400

4 22 2y x x

First derivative (slope) is zero

at:

0, 1, 1x

34 4dy x xdx