Section 2.6 1. Use the Generalized Power Rule – Chain Rule – to find the derivative of f (x) = (x 2 + 1) 3 f ( x) = (x2 +1) 3 f ′( x) = 3(x2 + 1) 2 (2x)

Section 2.6

1. Use the Generalized Power Rule – Chain Rule – to find the derivative of f (x) = (x 2 + 1) 3

f ( x) = (x2 +1)3

f ′( x) = 3(x2 + 1)2(2x) = 6x(x2 + 1)2

2. Use the Generalized Power Rule – Chain Rule – to find the derivative of f (x) = (3x 2 – 5x + 2) 4

f (x) = (3x2 − 5x + 2)4

f ′( x) = 4(3x2 − 5x + 2)3 (6x − 5)

3f (x) 9x 1

3. Use the Generalized Power Rule – Chain Rule – to find the derivative of

133f (x) 9x 1 (9x 1)

2 23 3

1f '(x) (9x 1) (9) 3(9x 1)

3

4. Use the Generalized Power Rule – Chain Rule – to find the derivative of f (x) = (4 - x 2 ) 4

y = (4 − x2 )4

y ′ = 4(4 − x2 )3(−2x) = −8x(4 − x2)3

5. Use the Generalized Power Rule – Chain Rule – to find the derivative of f (x) = x 4 + (1 – x ) 4

y = x 4 + (1 − x)4

y ′ = 4x3 + 4(1 − x)3(−1) = 4x 3 − 4(1− x)3

23

1f (x)

(9x 1)


2 3

23

1f (x) (9x 1)

(9x 1)

5 3 5 32f '(x) (9x 1) (9) 6(9x 1)

3

7. Use the Generalized Power Rule – Chain Rule – to find the derivative of f (x) = [ (x 2 + 1 ) 3 + x] 3

f ( x) = [(x2 +1)3 + x]3

f ′( x) = 3[( x2 + 1)3 + x]2 [3(x2 + 1)2 (2x) +1]

= 3[( x2 + 1)3 + x]2 [6x(x2 + 1)2 + 1]

8. Use the Generalized Power Rule – Chain Rule – to find the derivative of f (x) = (2x + 1) 3 (2x – 1) 4

f ( x) = (2x + 1)3(2x − 1)4

f ′( x) = (2x + 1)3[4(2x − 1)3(2)] + (2x − 1)4 3(2x + 1)2 (2)

Product Rule and Chain Rule combined.

f ( x) = 8(2x + 1)3(2x −1)3 + 6(2x + 1)2(2x −1)4


3x 1

f (x)x 1

2

2

x 1 (x 1)(1) (x 1)(1)f '(x) 3

x 1 (x 1)

Chain RuleQuotient Rule

2 2

2 2

x 1 2 6(x 1)f '(x) 3

x 1 (x 1) (x 1)

10. BUSINESS: Cost - A company’s cost function is given below in dollars, where x is the number of units. Find the marginal cost function and evaluate it at 20.

900x4)x(C 2

Since you are trying to calculate a derivative at a point you may use your calculator. Graph the original cost function and go to the derivative menu.

The marginal cost function is the derivative of the function C (x)

2 1 2C(x) (4x 900)

2 1 2 2 1 21C'(x) (4x 900) (8x) 4x(4x 900)

2

2 1 2C'(20) 4(20)[4(20) 900] 1.6

11. SOCIOLOGY: Income Status – A study estimated how a person’s social status (rated on a scale where 100 indicates the status of a college graduate) depends upon income. Based on this study, with a an income of x thousand dollars, a person’s status is given by S (x) = 17.5 (x – 1) 0.53. Find S ’ (25) and interpret your answer.

S(x) = 17.5(x −1)0.53

S ′ (x) = 9.275(x −1) −0.47

S ′ (25) = 9.275(25 − 1) −0.47 ≈ 2.08

At an income of $25,000 social status increases by about 2.08 units per additional $1000 ofincome.


12. BIOMEDICAL: Drug Sensitivity – The strength of a patient’s reaction to a dose of x milligrams of a certain drug is R (x) = 4x (11 + 0.5 x) 0.5 for 0 ≤ x ≤ 140. The derivative R‘ (x) is called the sensitivity to the drug. Find R‘ (50), the sensitivity to a dose of 50 milligrams of the drug.

5.0)x5.011(x4)x(R

0.5 0.5R'(x) (4x)0.5(11 0.5x) (0.5) 4(11 0.5x)

0.5 0.5x(11 0.5x) 4(11 0.5x)

0.5 0.5 50R(50) 50(11 25) 4(11 25) 4 6 32.33

6


13. ENVIRONMENTAL SCIENCE: Pollution – The carbon monoxide level in a city is predicted to be 0.02 x 2/3 + 1 ppm (parts per million), where x is the population in thousands. In t years the population of the city is predicted to be x (t) 12 + 2t thousand people. Therefore, in t years the carbon monoxide level will be

P 9t) = 0.02 (12 + 2t) 3/2 + 1 ppmFind P ‘ (2), the rate at which carbon monoxide pollution will be increasing in 2

years.

P(t ) = 0.02(12 + 2t)3/2 + 1

P′ (t ) = 0.03(12 + 2t ) 1/2 (2) = 0.06(12 + 2t) 1/2

P′ (2) = 0.06[12 + 2(2)]1/2 = 0.24


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Section 2.6 1. Use the Generalized Power Rule – Chain Rule – to find the derivative of f (x) = (x 2 + 1) 3 f ( x) = (x2 +1) 3 f ′( x) = 3(x2 + 1) 2 (2x)