Section 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) = (3x 2 − 5x + 2) 4 f ′( x) = 4(3x 2 − 5x + 2) 3 (6x − 5)

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

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

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

7. Use the Generalized Power Rule – Chain Rule – to find the derivative of f (x) = [ (x ) 3 + x] 3 f ( x) = [(x 2 +1) 3 + x] 3 f ′( x) = 3[( x 2 + 1) 3 + x] 2 [3(x 2 + 1) 2 (2x) +1] = 3[( x 2 + 1) 3 + x] 2 [6x(x 2 + 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

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

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. 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)

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) 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 of income. 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.

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 ( 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. 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.

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) t thousand people. Therefore, in t years the carbon monoxide level will be P 9t) = 0.02 (12 + 2t) 3/2 + 1 ppm Find 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 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.