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§3.5 Derivatives of Logarithmic and Exponential Functions.

§3.5 Derivatives of Logarithmic and Exponential Functions. The student will learn about:. the derivative of ln x and the ln f (x) ,. the derivative of ln x and the ln f (x) , the derivative of e x and e f (x) and,. applications. Derivative Formula for ln x.

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§3.5 Derivatives of Logarithmic and Exponential Functions.

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  1. §3.5 Derivatives of Logarithmic and Exponential Functions. The student will learn about: the derivative of ln x and the ln f (x), the derivative of ln x and the ln f (x), the derivative of e x and e f (x) and, applications.

  2. Derivative Formula for ln x. The above derivative can be combined with the power rule, product rule, quotient rule, and chain rule to find more complicated derivatives

  3. Examples. f (x) = 5 ln x. f ‘ (x) = (5)(1/x) = 5/x f (x) = x5 ln x. Note: We need the product rule. f ‘ (x) = (x 5 )(1/x) + (ln x)(5x 4 ) = x 4 + (ln x)(5x 4)

  4. OR Derivative Formula for ln f (x). We just learned that What if instead of x we had an ugly function? The above derivative can be combined with the power rule, product rule, quotient rule, and chain rule to find more complicated derivatives

  5. Examples. f (x) = ln (x 4 + 5) f ‘ (x) = f ‘ (x) = f (x) = 4 ln √x = 4 ln x 1/2

  6. Examples. f (x) = (5 – 3 ln x) 4 . f ‘ (x) = 4 (5 – 3 ln x) 3 f ‘ (x) = = 4 (5 – 3 ln x) 3

  7. Derivative Formulas for ex. The above derivative can be combined with the power rule, product rule, quotient rule, and chain rule to find more complicated derivatives

  8. Examples. Find derivatives for f (x) = 3 e x. f ‘ (x) = 3 e x . f (x) = x 4 e x Hint, use the product rule. f ‘ (x) = x 4 e x + ex 4x 3

  9. Derivative Formulas for e f (x). We just learned that What if instead of x we had an ugly function? OR The above derivative can be combined with the power rule, product rule, quotient rule, and chain rule to find more complicated derivatives

  10. Example.

  11. General Derivative Rules Power Rule General Power Rule Exponential Rule General Exponential Derivative Rule Log Rule General Log Derivative Rule

  12. Maximizing Consumer Expenditure The amount of a commodity that consumers will buy depends on the price of the commodity. For a commodity whose price is p, let the consumer demand be given by a function D(p). Multiplying the number of units D(p) by the price p gives the total consumer expenditure for the commodity.

  13. 0 ≤ x ≤ 15 and 0 ≤ y ≤ 6,000 ExampleConsumer Demand and Expenditure. The consumer expenditure, is E (p) = p · D (p), where D is the demand function. Let consumer demand be D (p) = 8000 e – 0.05 p Graph this on your calculator and see if it makes sense.

  14. 0 ≤ x ≤ 30 and 0 ≤ y ≤ 65,000 Consumer Demand and Expenditure. Continued The consumer expenditure, is E (p) = p · D (p), where D is the demand function. Let consumer demand be D (p) = 8000 e – 0.05 p Maximize the consumer expenditure. Consumer expenditure E (p) = p 8000 e – 0.05 p Use your calculator to maximize this. E (20) = $58,860.71

  15. Summary. • The derivative of f (x) = ln x is f ' (x) = 1/x. • The derivative of f (x) = ln u is f ' (x) = (1/u) u'. • The derivative of f (x) = ex is f ' (x) = ex. • The derivative of f (x) = eu is f ' (x) = eu u'. • We did an application involving consumer expenditure.

  16. ASSIGNMENT §3.5 on my website.

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