Derivatives of Exponential and Logarithmic Functions Michael Freeze - - PowerPoint PPT Presentation

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Derivatives of Exponential and Logarithmic Functions Michael Freeze MAT 151 UNC Wilmington Summer 2013 1 / 23 Section 4.4 :: Derivatives of Exponential Functions 2 / 23 Derivative of e x d dx ( e x ) = e x Example Find the derivative of y

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Derivatives of Exponential and Logarithmic Functions

Michael Freeze

MAT 151 UNC Wilmington

Summer 2013

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Section 4.4 :: Derivatives of Exponential Functions

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Derivative of ex d dx (ex) = ex

Example

Find the derivative of y = 5ex.

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Derivative of ex d dx (ex) = ex

Example

Find the derivative of y = ex+7.

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Derivative of ex d dx (ex) = ex

Example

Find the derivative of y = x ex.

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Derivative of ex d dx (ex) = ex

Example

Find the derivative of y =

ex x+1.

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Derivative of ax For any positive constant a = 1, d dx (ax) = (ln a) ax

Example

Find the derivative of f (x) = 2x.

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Chain rule for exponential functions d dx

eg(x)

= eg(x) · g ′(x)

Example

Find the derivative of y = e3x.

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Chain rule for exponential functions d dx

eg(x)

= eg(x) · g ′(x)

Example

Find the derivative of y = ex2−x.

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SLIDE 10

Chain rule for exponential functions d dx

eg(x)

= eg(x) · g ′(x)

Example

Find the derivative of y = −3e−x2+3.

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Chain rule for exponential functions d dx

eg(x)

= eg(x) · g ′(x)

Example

Find the derivative of y = (x2 + x)e2x−1.

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Pollution Concentration

The concentration of pollutants (in grams per liter) in the east fork of the Big Weasel River is approximated by P(x) = 0.04e−4x, where x is the number of miles downstream from a paper mill that the measurement is taken.

Find the following values. (a) The concentration of pollutants 0.5 mile downstream (b) The concentration of pollutants 1 mile downstream (c) The concentration of pollutants 2 miles downstream

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Pollution Concentration

The concentration of pollutants (in grams per liter) in the east fork of the Big Weasel River is approximated by P(x) = 0.04e−4x, where x is the number of miles downstream from a paper mill that the measurement is taken.

Find the rate of change of concentration with respect to dis- tance for the following distances. (d) 0.5 mile (e) 1 mile (f) 2 miles

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Section 4.5 :: Derivatives of Logarithmic Functions

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Derivative of ln x d dx [ln x] = 1 x

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Chain Rule for Logarithmic Functions d dx [ln g(x)] = g ′(x) g(x)

Example

Find the derivative of y = ln(8x).

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Chain Rule for Logarithmic Functions d dx [ln g(x)] = g ′(x) g(x)

Example

Find the derivative of y = ln(x2 − 3x + 1).

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Chain Rule for Logarithmic Functions d dx [ln g(x)] = g ′(x) g(x)

Example

Find the derivative of y = x ln x.

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Chain Rule for Logarithmic Functions d dx [ln g(x)] = g ′(x) g(x)

Example

Find the derivative of y = ln(√4x − 3).

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Chain Rule for Logarithmic Functions d dx [ln g(x)] = g ′(x) g(x)

Example

Find the derivative of y = ln

3x−1

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Logarithmic Differentiation Find the derivative of y = (x + 4)4 (3x − 1)2.

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Logarithmic Differentiation Find the derivative of y = √3x − 1 (x2 + 2x)3.

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Field Metabolic Rate

The field metabolic rate (FMR), or the total energy expenditure per day in excess of growth, can be calculated for pronghorn fawns using Nagy’s formula, F(x) = 0.774 + 0.727 log x, where x is the mass (in grams) of the fawn and F(x) is the energy expenditure (in kJ/day). Source: Animal Behavior. (a) Determine the total energy expenditure per day in excess of growth for a pronghorn fawn that weighs 25,000 g. (b) Find F ′(125, 000) and interpret the result.

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