The method of logarithmic differentiation , calculus, uses the properties of logarithmic functions to differentiate complicated functions and functions where the usual formulas of Differentiation do not apply. Several examples with detailed solutions are presented.

Example 1

Solution to Example 1

• We first note that there is no formula that can be used to differentiate directly this function. The first derivative may be calculated by first taking the natural logarithm of both sides of y = x ^{sin x}.
  ln y = ln [ x ^{sin x} ]
  
• Use logarithm properties to rewrite the above equation as follows
  ln y = sin x ln x
  
• We now differentiate both sides with respect to x, using the chain rule on the left side and the product rule formula for differentiation on the right side.
  y ' / y = cos x ln x + sin x (1/x)
  
• Multiply both sides by y to obtain
  y ' = [ cos x ln x + (1/x) sin x ] y
  y ' = [ cos x ln x + (1/x) sin x ] x ^{sin x}

Example 2

Solution to Example 2

• We take the logarithms of both sides
  ln y = ln x + ln e ^{(-x 2)}
  
• Simplify the term ln e ^{(-x 2)}
  ln y = ln x - x ²
  
• Differentiate both sides with respect to x.
  y ' / y = 1 / x + 2 x
  
• Multiply all terms by y
  y ' = [ 1 / x + 2 x ] y
  y ' = [ 1 / x + 2 x ] x e ^{(-x 2)}
  y ' = [ 1 + 2 x ² ] e ^{(-x 2)}

Example 3

Solution to Example 3

• We take the logarithms of both sides
  ln y = ln 3 + ln x ² + ln e ^-x
  
• Simplify the term ln e ^-x
  ln y = ln 3 + 2 ln x - x
  
• Differentiate both sides with respect to x.
  y ' / y = 0 + 2 / x - 1
  
• Multiply all terms by y
  y ' = [ 2 / x - 1 ] y
  y ' = [ 2 / x - 1 ] 3 x ² e ^-x
  y ' = [ 6 x - 3 x ² ] e ^-x
  y ' = 3 x [ 2 - x ] e ^-x
  NOTE: As an exercise, use the usual formula of differentiation to differentiate the above function and compare results.

Example 4

Solution to Example 4

• Take the logarithms of both sides and expand expressions obtained
  ln y = 2 ln (1 - x) + 4 ln (x + 1)
  
• Differentiate both sides with respect to x.
  y ' / y = - 2 / (1 - x) + 4 / (x + 1)
  
• Multiply all terms by y
  y ' = [ - 2 / (1 - x) + 4 / (x + 1) ] y
  y ' = [ - 2 / (1 - x) + 4 / (x + 1) ] (1 - x) ² (x + 1) ⁴
  y ' = - 2 (1 - x)(x + 1) ⁴ + 4 (x + 1) ³(1 - x) ²
  y ' = 2 (1 - x)(x + 1) ³ (3x - 1)
  NOTE: Use the usual formula of differentiation to differentiate the above function and compare results.

Example 5

Solution to Example 5

• Take the logarithms of both sides
  ln y = ln (tan x) - ln e ^x
  
• Simplify ln e ^x.
  ln y = ln (tan x) - x
  
• Differentiate both sides with respect to x
  y ' / y = sec x / tan x - 1
  
• Multiply all terms by y
  y ' = [ sec x / tan x - 1 ] y
  y ' = [ sec ²x / tan x -1 ] tan x / e ^x
  y ' = [ sec ²x - tan x ] / e ^x
  NOTE: Use the usual formula of differentiation to differentiate the above function and compare results.

Example 6

Solution to Example 6

• Take the logarithms of both sides and expand the expressions obtained
  ln y = ln (x - 2) + ln (x + 4) - ln (x + 1) - ln (x + 5)
  
• Differentiate both sides with respect to x
  y ' / y = 1 / (x - 2) + 1 / (x + 4) - 1 / (x + 1) - 1 / (x + 5)
  
• Multiply all terms by y
  y ' = [ 1 / (x - 2) + 1 / (x + 4) - 1 / (x + 1) - 1 / (x + 5) ] y
  
• Substitute y by its formula to obtain
  y ' = 2 [ 2x ² + 13 x + 29] / [ (x + 1) ² (x + 5) ² ]
  NOTE: Use the usual formula of differentiation to differentiate the above function and compare results.

Example 7

Solution to Example 7

• Take the logarithms of both sides and expand the expressions obtained
  ln y = ln u + ln v
  
• Differentiate both sides with respect to x
  y ' / y = u ' / u + v ' / v
  
• Multiply all terms by y and simplify to obtain y ' = [ u ' / u + v ' / v ] y
  y ' = [ u ' / u + v ' / v ] u v
  y ' = u ' v + v ' u
  NOTE: The result obtained is the well known product rule of differentiation.

Example 8

Solution to Example 8

• Take the logarithms of both sides and expand the expressions obtained using the logarithm properties
  ln y = ln u - ln v
  
• Differentiate both sides with respect to x using the differentiation rule of the logarithm of a function
  y ' / y = u ' / u - v ' / v
  
• Multiply all terms by y and simplify to obtain
  y ' = [ u ' / u - v ' / v ] y
  y ' = [ u ' / u - v ' / v ] [ u / v ]
  y ' = u ' / v - v ' u / [ v ² ]
  y ' = [ u ' v - v ' u ] / [ v ² ]
  NOTE: The result obtained is the well known quotient rule of differentiation of functions.

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