Derivatives of Exponential Functions

1
Derivatives of Exponential Functions

• Section
• 7.6

2
Proper Lingo ?

• Finding a derivative differentiation
• The derivative that we find differential
  equation

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5 new derivatives

• f(x) sin x h(x) ln x
• f(x) cos x h(x) 1/x
• g(x) cos x j(x) abx
• g(x) -sin x j(x) (ln b)abx
• k(x) ex
• k(x) ex

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Example 1

• The number of cell phone subscribers (in mil)
  f(x) from 2002 2007 is closely modeled by f(x)
  125.5(1.129)x, where x is the number of years
  after Jan. 1, 2001. What was the instantaneously
  rate of change of the number of subscribers on
  January 1, 2005?
• f(x) ln 1.129 125.5 (1.129)x
• f(4) ln 1.129 125.5 (1.129)4
• .12133 203.9
• 24.74 mil/year

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Example 2

• F(x) eax where a is a constant. What is f(x)
  ?
• Rewrite f(x) (ea)x
• so f(x) ln (ea) (ea)x
• f(x) a (ea)x
• f(x) aeax

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Example 3

• Equation r(x) 2x x2
• Find the slope of the tangent line at x 3.
• f(x) ln(2) 2x 2x
• f(3) ln(2) 8 6
• Find an equation for this tangent line.
• Find a point on the line ? (3, 17)
• y 17 (ln(2) 8 6) (x 3)
• y 11.545x 17.6355

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Homework

• Pages 454 455
• 1, 7- 12,
• 14, 15