Derivatives - Logarithmic Differentiation

• Logarithmic differentiation is a technique used to differentiate functions that are products, quotients, or powers involving logarithmic functions.
• It is particularly useful when dealing with complicated functions that cannot be easily differentiated using basic differentiation techniques.
• This method involves taking the natural logarithm of both sides of an equation and then differentiating implicitly.
• Let’s see how this technique can be applied to various types of functions.
• Example: Differentiate the function y = (2x^3 + 3x^2 + 4x + 5) / (e^x)

Logarithmic Differentiation Technique

1. Start by taking the natural logarithm of both sides of the equation.

• Example: ln(y) = ln((2x^3 + 3x^2 + 4x + 5) / (e^x))

2. Apply the logarithm properties to simplify the equation.

• Example: ln(y) = ln(2x^3 + 3x^2 + 4x + 5) - ln(e^x) = ln(2x^3 + 3x^2 + 4x + 5) - x

3. Differentiate both sides of the equation implicitly.

• Example: d/dx(ln(y)) = d/dx(ln(2x^3 + 3x^2 + 4x + 5) - x)

4. Use the chain rule to differentiate the natural logarithm of y.

• Example: (1/y) * dy/dx = (1/(2x^3 + 3x^2 + 4x + 5)) * (6x^2 + 6x + 4) - 1

5. Solve for dy/dx, which is the derivative of the original function.

• Example: dy/dx = y * [(6x^2 + 6x + 4) / (2x^3 + 3x^2 + 4x + 5)] - y

Logarithmic Differentiation Examples

1. Example: Differentiate the function y = x^x

• Apply logarithmic differentiation technique to simplify the expression: ln(y) = ln(x^x) ln(y) = x * ln(x)
• Differentiate implicitly: (1/y) * dy/dx = 1 * ln(x) + x * (1/x) dy/dx = y * [ln(x) + 1]

2. Example: Differentiate the function y = (sin(x))^cos(x)

• Apply logarithmic differentiation technique: ln(y) = ln((sin(x))^cos(x)) ln(y) = cos(x) * ln(sin(x))
• Differentiate implicitly: (1/y) * dy/dx = -sin(x) * ln(sin(x)) + cos(x) * (1/sin(x)) * cos(x) dy/dx = y * (-sin(x) * ln(sin(x)) + cos^2(x) / sin(x))

3. Example: Differentiate the function y = x^(x^x)

• Apply logarithmic differentiation technique: ln(y) = ln(x^(x^x)) ln(y) = x^x * ln(x)
• Differentiate implicitly: (1/y) * dy/dx = (x^x) * (1/x) * ln(x) + (x^x) * (ln(x) + x * (1/x)) dy/dx = y * (x^x / x * ln(x) + x^x * (ln(x) + 1)) These were examples of logarithmic differentiation applied to different types of functions. Practice using this technique to solve more complex functions.

Slide 11: Logarithmic Differentiation Technique (contd.)

• Example: Differentiate the function y = x^2 / sqrt(x+1)
  • Apply logarithmic differentiation technique:
    • ln(y) = ln(x^2 / sqrt(x+1))
    • ln(y) = 2ln(x) - (1/2)ln(x+1)
  • Differentiate implicitly:
    • (1/y) * dy/dx = 2 * (1/x) - (1/2) * (1/(x+1))
    • dy/dx = y * [2/x - 1/(2(x+1))]
• Example: Differentiate the function y = (x^3 + 1)^x / e^(2x)
  • Apply logarithmic differentiation technique:
    • ln(y) = ln((x^3 + 1)^x / e^(2x))
    • ln(y) = x * ln(x^3 + 1) - 2x
  • Differentiate implicitly:
    • (1/y) * dy/dx = ln(x^3 + 1) + x * (3x^2 / (x^3 + 1)) - 2
    • dy/dx = y * [ln(x^3 + 1) + (3x^3 / (x^3 + 1)) - 2]
• Example: Differentiate the function y = ln(sqrt(x)) / cos(x)
  • Apply logarithmic differentiation technique:
    • ln(y) = ln(ln(sqrt(x)) / cos(x))
    • ln(y) = ln(ln(sqrt(x))) - ln(cos(x))
  • Differentiate implicitly:
    • (1/y) * dy/dx = (1/ln(sqrt(x))) * (1/(2sqrt(x))) - (-tan(x))
    • dy/dx = y * [(1/(ln(sqrt(x)))) * (1/(2sqrt(x))) + tan(x)]

Slide 12: Logarithmic Differentiation Examples

• Example: Differentiate the function y = e^x * ln(e^x) / cos^2(x)
  • Apply logarithmic differentiation technique:
    • ln(y) = ln(e^x * ln(e^x)) - 2ln(cos(x))
    • ln(y) = x + ln(ln(e^x)) - 2ln(cos(x))
  • Differentiate implicitly:
    • (1/y) * dy/dx = 1 + (1/ln(e^x)) * (1/e^x) - 2 * (-tan(x)/cos(x))
    • dy/dx = y * [1 + (1/(ln(e^x))) * (1/e^x) + 2 * (tan(x)/cos(x))]
• Example: Differentiate the function y = (log(x))^2 * sqrt(log(x))
  • Apply logarithmic differentiation technique:
    • ln(y) = ln((log(x))^2 * sqrt(log(x)))
    • ln(y) = 2ln(log(x)) + (1/2)ln(log(x))
  • Differentiate implicitly:
    • (1/y) * dy/dx = (2 * (1/(log(x))) * (1/x)) + (1/2) * (1/(x * log(x)))
    • dy/dx = y * [2/(x * log(x)) + 1/(2x * log(x))]
• Example: Differentiate the function y = (e^x * ln(x))^2 / x^3
  • Apply logarithmic differentiation technique:
    • ln(y) = ln((e^x * ln(x))^2 / x^3)
    • ln(y) = 2ln(e^x * ln(x)) - 3ln(x)
  • Differentiate implicitly:
    • (1/y) * dy/dx = (2 * (1/(e^x * ln(x))) * (e^x * ln(x) + e^x/x)) - (3 * (1/x))
    • dy/dx = y * [2 * (e^x + 1/x) - (3/x)]

Add more slides as needed Derivatives - Logarithmic Differentiation

Slide 21:

• Example: Differentiate the function y = e^(2x) * ln(3x)
  • Apply logarithmic differentiation technique:
    • ln(y) = ln(e^(2x) * ln(3x))
    • ln(y) = 2x + ln(ln(3x))
  • Differentiate implicitly:
    • (1/y) * dy/dx = 2 + (1/(ln(3x))) * (1/(3x)) * 3
    • dy/dx = y * [2 + 1/(3x * ln(3x))]

Slide 22:

• Example: Differentiate the function y = (log₂(x))^2 / x^3
  • Apply logarithmic differentiation technique:
    • ln(y) = ln((log₂(x))^2 / x^3)
    • ln(y) = 2ln(log₂(x)) - 3ln(x)
  • Differentiate implicitly:
    • (1/y) * dy/dx = (2 * (1/(log₂(x))) * (1/(x * ln(2)))) - (3 * (1/x))
    • dy/dx = y * [2/(x * ln(2) * log₂(x)) - 3/x]

Slide 23:

• Example: Differentiate the function y = (x^n) / (logₐ(x))
  • Apply logarithmic differentiation technique:
    • ln(y) = ln((x^n) / (logₐ(x)))
    • ln(y) = n * ln(x) - ln(logₐ(x))
  • Differentiate implicitly:
    • (1/y) * dy/dx = (n * (1/x)) - (1/(x * ln(a) * logₐ(x)))
    • dy/dx = y * [n/x - 1/(x * ln(a) * logₐ(x))]

Slide 24:

• Example: Differentiate the function y = (2^(2x) - 3^x) / (log(x)^2)
  • Apply logarithmic differentiation technique:
    • ln(y) = ln((2^(2x) - 3^x) / (log(x)^2))
    • ln(y) = ln((2^(2x) - 3^x)) - 2ln(log(x))
  • Differentiate implicitly:
    • (1/y) * dy/dx = (2^(2x) * ln(2) - 3^x * ln(3)) - (2 * (1/x))
    • dy/dx = y * [(2^(2x) * ln(2) - 3^x * ln(3)) / (x * log(x))]

Slide 25:

• Example: Differentiate the function y = (e^(x^2) - ln(x^3)) / x
  • Apply logarithmic differentiation technique:
    • ln(y) = ln((e^(x^2) - ln(x^3)) / x)
    • ln(y) = ln((e^(x^2) - ln(x^3))) - ln(x)
  • Differentiate implicitly:
    • (1/y) * dy/dx = (2x * e^(x^2) - (3/x^3)) - (1/x)
    • dy/dx = y * [(2x * e^(x^2) - (3/x^3)) / x]

Slide 26:

• Example: Differentiate the function y = ((ln(x))^2 - e^x * cos(x)) / x^2
  • Apply logarithmic differentiation technique:
    • ln(y) = ln(((ln(x))^2 - e^x * cos(x)) / x^2)
    • ln(y) = 2ln(ln(x)) - ln(e^x * cos(x)) - 2ln(x)
  • Differentiate implicitly:
    • (1/y) * dy/dx = (2 * (1/(ln(x))) * (1/x)) - (e^x * cos(x) + e^x * sin(x)) - (2 * (1/x))
    • dy/dx = y * [2/(x * ln(x)) - (e^x * cos(x) + e^x * sin(x)) / x]

Slide 27:

• Example: Differentiate the function y = (2^(x^2) - tan(x)) / (log(x) * cos(x))
  • Apply logarithmic differentiation technique:
    • ln(y) = ln((2^(x^2) - tan(x)) / (log(x) * cos(x)))
    • ln(y) = ln((2^(x^2) - tan(x))) - ln(log(x) * cos(x))
  • Differentiate implicitly:
    • (1/y) * dy/dx = (2^(x^2) * ln(2) - sec^2(x) - (1/(x * ln(x))) - (tan(x) * cot(x)))
    • dy/dx = y * [(2^(x^2) * ln(2) - sec^2(x) - (1/(x * ln(x))) - (tan(x) * cot(x)))]

Slide 28:

• Example: Differentiate the function y = (log₂(x) - ln(x)) / sin(x)
  • Apply logarithmic differentiation technique:
    • ln(y) = ln((log₂(x) - ln(x)) / sin(x))
    • ln(y) = ln((log₂(x) - ln(x))) - ln(sin(x))
  • Differentiate implicitly:
    • (1/y) * dy/dx = ((1/(x * ln(2))) - (1/x)) - (1/sin(x)) * cos(x)
    • dy/dx = y * [(1/(x * ln(2))) - (1/x) - (cos(x)/sin(x))]

Slide 29:

• Example: Differentiate the function y = (e^x * ln(x)) / (cos(x) * sin(x))
  • Apply logarithmic differentiation technique:
    • ln(y) = ln((e^x * ln(x)) / (cos(x) * sin(x)))
    • ln(y) = ln(e^x * ln(x)) - ln(cos(x) * sin(x))
  • Differentiate implicitly:
    • (1/y) * dy/dx = (e^x * (1/x) + (1/x)) - ((cos(x) * sin(x)) + (cos(x) * cos(x) - sin(x) * sin(x)) / (cos(x) * sin(x)))
    • dy/dx = y * [(e^x * (1/x) + (1/x)) - (cos(x) * cos(x) + sin(x) * sin(x)) / (cos(x) * sin(x)))]

Slide 30:

• Example: Differentiate the function y = (log(x) * sin(x)) / (e^x * cos(x))
  • Apply logarithmic differentiation technique:
    • ln(y) = ln((log(x) * sin(x)) / (e^x * cos(x)))
    • ln(y) = ln(log(x) * sin(x)) - ln(e^x * cos(x))
  • Differentiate implicitly:
    • (1/y) * dy/dx = ((1/x) * sin(x) + log(x) * cos(x)) - ((e^x * sin(x) + e^x * cos(x) / (e^x * cos(x)))
    • dy/dx = y * [((1/x) * sin(x) + log(x) * cos(x)) - (e^x * sin(x) + e^x * cos(x) / (e^x * cos(x)))]