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Problem-solving strategy: using logarithmic differentiation

  1. To differentiate y = h ( x ) using logarithmic differentiation, take the natural logarithm of both sides of the equation to obtain ln y = ln ( h ( x ) ) .
  2. Use properties of logarithms to expand ln ( h ( x ) ) as much as possible.
  3. Differentiate both sides of the equation. On the left we will have 1 y d y d x .
  4. Multiply both sides of the equation by y to solve for d y d x .
  5. Replace y by h ( x ) .

Using logarithmic differentiation

Find the derivative of y = ( 2 x 4 + 1 ) tan x .

Use logarithmic differentiation to find this derivative.

ln y = ln ( 2 x 4 + 1 ) tan x Step 1. Take the natural logarithm of both sides. ln y = tan x ln ( 2 x 4 + 1 ) Step 2. Expand using properties of logarithms. 1 y d y d x = sec 2 x ln ( 2 x 4 + 1 ) + 8 x 3 2 x 4 + 1 · tan x Step 3. Differentiate both sides. Use the product rule on the right. d y d x = y · ( sec 2 x ln ( 2 x 4 + 1 ) + 8 x 3 2 x 4 + 1 · tan x ) Step 4. Multiply by y on both sides. d y d x = ( 2 x 4 + 1 ) tan x ( sec 2 x ln ( 2 x 4 + 1 ) + 8 x 3 2 x 4 + 1 · tan x ) Step 5. Substitute y = ( 2 x 4 + 1 ) tan x .
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Using logarithmic differentiation

Find the derivative of y = x 2 x + 1 e x sin 3 x .

This problem really makes use of the properties of logarithms and the differentiation rules given in this chapter.

ln y = ln x 2 x + 1 e x sin 3 x Step 1. Take the natural logarithm of both sides. ln y = ln x + 1 2 ln ( 2 x + 1 ) x ln e 3 ln sin x Step 2. Expand using properties of logarithms. 1 y d y d x = 1 x + 1 2 x + 1 1 3 cos x sin x Step 3. Differentiate both sides. d y d x = y ( 1 x + 1 2 x + 1 1 3 cot x ) Step 4. Multiply by y on both sides. d y d x = x 2 x + 1 e x sin 3 x ( 1 x + 1 2 x + 1 1 3 cot x ) Step 5. Substitute y = x 2 x + 1 e x sin 3 x .
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Extending the power rule

Find the derivative of y = x r where r is an arbitrary real number.

The process is the same as in [link] , though with fewer complications.

ln y = ln x r Step 1. Take the natural logarithm of both sides. ln y = r ln x Step 2. Expand using properties of logarithms. 1 y d y d x = r 1 x Step 3. Differentiate both sides. d y d x = y r x Step 4. Multiply by y on both sides. d y d x = x r r x Step 5. Substitute y = x r . d y d x = r x r 1 Simplify.
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Use logarithmic differentiation to find the derivative of y = x x .

d y d x = x x ( 1 + ln x )

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Find the derivative of y = ( tan x ) π .

y = π ( tan x ) π 1 sec 2 x

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Key concepts

  • On the basis of the assumption that the exponential function y = b x , b > 0 is continuous everywhere and differentiable at 0, this function is differentiable everywhere and there is a formula for its derivative.
  • We can use a formula to find the derivative of y = ln x , and the relationship log b x = ln x ln b allows us to extend our differentiation formulas to include logarithms with arbitrary bases.
  • Logarithmic differentiation allows us to differentiate functions of the form y = g ( x ) f ( x ) or very complex functions by taking the natural logarithm of both sides and exploiting the properties of logarithms before differentiating.

Key equations

  • Derivative of the natural exponential function
    d d x ( e g ( x ) ) = e g ( x ) g ( x )
  • Derivative of the natural logarithmic function
    d d x ( ln g ( x ) ) = 1 g ( x ) g ( x )
  • Derivative of the general exponential function
    d d x ( b g ( x ) ) = b g ( x ) g ( x ) ln b
  • Derivative of the general logarithmic function
    d d x ( log b g ( x ) ) = g ( x ) g ( x ) ln b

For the following exercises, find f ( x ) for each function.

f ( x ) = x 2 e x

2 x e x + x 2 e x

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f ( x ) = e x 3 ln x

e x 3 ln x ( 3 x 2 ln x + x 2 )

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f ( x ) = e x e x e x + e x

4 ( e x + e x ) 2

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Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
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2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
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Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
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Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
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