11. Product Rule:

• The product rule states that the derivative of the product of two functions is the first function times the derivative of the second function, plus the second function times the derivative of the first function.
• In mathematical notation, if f(x) = u(x)v(x), then the derivative of f(x) with respect to x is given by f’(x) = u’(x)v(x) + u(x)v’(x).
• Example: Find the derivative of f(x) = x^2 * ln(x).
  • Solution: We use the product rule. Let u(x) = x^2 and v(x) = ln(x).
  • Then, u’(x) = 2x and v’(x) = 1/x.
  • Applying the product rule, f’(x) = 2x * ln(x) + x^2 * (1/x).
  • Simplifying, f’(x) = 2x * ln(x) + x.
• The product rule is essential in finding derivatives of functions that involve multiplication.

12. Quotient Rule:

• The quotient rule states that the derivative of the quotient of two functions is given by the denominator times the derivative of the numerator, minus the numerator times the derivative of the denominator, divided by the square of the denominator.
• In mathematical notation, if f(x) = u(x)/v(x), then the derivative of f(x) with respect to x is given by f’(x) = (v(x)u’(x) - u(x)v’(x))/(v(x))^2.
• Example: Find the derivative of f(x) = (2x^3 + 1)/(x^2).
  • Solution: We use the quotient rule. Let u(x) = 2x^3 + 1 and v(x) = x^2.
  • Then, u’(x) = 6x^2 and v’(x) = 2x.
  • Applying the quotient rule, f’(x) = (x^2 * 6x^2 - (2x^3 + 1) * 2x)/(x^2)^2.
  • Simplifying, f’(x) = (6x^4 - 4x^4 - 2x)/(x^4).
  • Further simplification yields f’(x) = (2x^4 - 2x)/(x^4).

13. Power Rule:

• The power rule states that the derivative of a function raised to a constant power is equal to the constant times the function raised to the power minus 1, times the derivative of the function.
• In mathematical notation, if f(x) = u(x)^n, where n is a constant, then the derivative of f(x) with respect to x is given by f’(x) = n(u(x))^(n-1) * u’(x).
• Example: Find the derivative of f(x) = x^5.
  • Solution: We use the power rule. Let u(x) = x and n = 5.
  • Then, u’(x) = 1.
  • Applying the power rule, f’(x) = 5(x^4) * 1.
  • Simplifying, f’(x) = 5x^4.
• The power rule is valuable in finding derivatives of functions raised to a constant power.

14. Derivative of ln(x):

• The derivative of the natural logarithm function, ln(x), where x > 0, is equal to 1/x.
• In mathematical notation, if f(x) = ln(x), then the derivative of f(x) with respect to x is given by f’(x) = 1/x.
• Example: Find the derivative of f(x) = ln(x^2).
  • Solution: We can rewrite f(x) as f(x) = 2ln(x).
  • Using the derivative of ln(x), f’(x) = 2 * (1/x).
  • Simplifying, f’(x) = 2/x.

15. Derivative of ln(u), where u is a function of x:

• To find the derivative of ln(u), where u is a function of x, we use the chain rule.
• The chain rule states that if f(u) is a composite function, then the derivative of f(u) with respect to x is given by f’(u) * u’(x).
• Example: Find the derivative of f(x) = ln(x^2 + 1).
  • Solution: Let u = x^2 + 1.
  • Using the chain rule, f’(x) = (1/u) * (2x).
  • Simplifying, f’(x) = (2x)/(x^2 + 1).

16. Example of finding derivatives of logarithmic functions:

• Example 1: Find the derivative of f(x) = ln(3x^2 + 2).
  • Solution: Using the chain rule, f’(x) = (1/(3x^2 + 2)) * (6x).
  • Simplifying, f’(x) = 6x/(3x^2 + 2).
• Example 2: Find the derivative of f(x) = ln(2x^3 - 4x).
  • Solution: Using the chain rule, f’(x) = (1/(2x^3 - 4x)) * (6x^2 - 4).
  • Simplifying, f’(x) = (6x^2 - 4)/(2x^3 - 4x).

17. The chain rule for logarithmic functions:

• The chain rule is a valuable technique for finding the derivative of composite functions, including logarithmic functions.
• The chain rule allows us to differentiate functions within functions.
• Example: Find the derivative of f(x) = ln(sin(x)).
  • Solution: Let u = sin(x).
  • Using the chain rule, f’(x) = (1/u) * cos(x).
  • Simplifying, f’(x) = cos(x)/sin(x) = cot(x).
• The chain rule extends the power of differentiation to more complex functions.

18. Application of logarithmic functions in calculus:

• Logarithmic functions find extensive applications in calculus, particularly in solving exponential growth and decay problems.
• They are used to model population growth, radioactive decay, compound interest, and many other phenomena.
• Logarithmic differentiation, which relies on logarithmic functions, is a powerful technique for finding derivatives when the function is expressed as the product, quotient, or power of other functions.
• The natural logarithm function, ln(x), is especially important in calculus as it helps simplify and solve various equations and integrals.

19. Summary of properties of logarithmic functions:

• Logarithmic functions have several essential properties, which are useful for various calculations and manipulations:
  1. Logarithm of a product: log(ab) = log(a) + log(b)
  2. Logarithm of a quotient: log(a/b) = log(a) - log(b)
  3. Logarithm of a power: log(a^b) = b * log(a)
  4. Logarithm of the base: log_b(b) = 1
  5. Logarithm of 1: log_b(1) = 0
  6. Change of base formula: log_a(x) = log_b(x)/log_b(a)
• These properties help simplify calculations, solve equations, and manipulate logarithmic functions efficiently.

Derivatives - Properties of Logarithmic Function

• Review of logarithmic function
• The natural logarithm function
• Basic properties of logarithmic function - Product rule - Quotient rule - Power rule
• Derivatives of logarithmic functions - Derivative of ln(x) - Derivative of ln(u), where u is a function of x

Product Rule

• The product rule states that the derivative of the product of two functions is the first function times the derivative of the second function, plus the second function times the derivative of the first function.
• In mathematical notation, if f(x) = u(x)v(x), then the derivative of f(x) with respect to x is given by f’(x) = u’(x)v(x) + u(x)v’(x).
• Example: Find the derivative of f(x) = x^2 * ln(x).
  • Solution: We use the product rule. Let u(x) = x^2 and v(x) = ln(x).
  • Then, u’(x) = 2x and v’(x) = 1/x.
  • Applying the product rule, f’(x) = 2x * ln(x) + x^2 * (1/x).
  • Simplifying, f’(x) = 2x * ln(x) + x.

Quotient Rule

• The quotient rule states that the derivative of the quotient of two functions is given by the denominator times the derivative of the numerator, minus the numerator times the derivative of the denominator, divided by the square of the denominator.
• In mathematical notation, if f(x) = u(x)/v(x), then the derivative of f(x) with respect to x is given by f’(x) = (v(x)u’(x) - u(x)v’(x))/(v(x))^2.
• Example: Find the derivative of f(x) = (2x^3 + 1)/(x^2).
  • Solution: We use the quotient rule. Let u(x) = 2x^3 + 1 and v(x) = x^2.
  • Then, u’(x) = 6x^2 and v’(x) = 2x.
  • Applying the quotient rule, f’(x) = (x^2 * 6x^2 - (2x^3 + 1) * 2x)/(x^2)^2.
  • Simplifying, f’(x) = (6x^4 - 4x^4 - 2x)/(x^4).
  • Further simplification yields f’(x) = (2x^4 - 2x)/(x^4).

Power Rule

• The power rule states that the derivative of a function raised to a constant power is equal to the constant times the function raised to the power minus 1, times the derivative of the function.
• In mathematical notation, if f(x) = u(x)^n, where n is a constant, then the derivative of f(x) with respect to x is given by f’(x) = n(u(x))^(n-1) * u’(x).
• Example: Find the derivative of f(x) = x^5.
  • Solution: We use the power rule. Let u(x) = x and n = 5.
  • Then, u’(x) = 1.
  • Applying the power rule, f’(x) = 5(x^4) * 1.
  • Simplifying, f’(x) = 5x^4.

Derivative of ln(x)

• The derivative of the natural logarithm function, ln(x), where x > 0, is equal to 1/x.
• In mathematical notation, if f(x) = ln(x), then the derivative of f(x) with respect to x is given by f’(x) = 1/x.
• Example: Find the derivative of f(x) = ln(x^2).
  • Solution: We can rewrite f(x) as f(x) = 2ln(x).
  • Using the derivative of ln(x), f’(x) = 2 * (1/x).
  • Simplifying, f’(x) = 2/x.

Derivative of ln(u), where u is a function of x

• To find the derivative of ln(u), where u is a function of x, we use the chain rule.
• The chain rule states that if f(u) is a composite function, then the derivative of f(u) with respect to x is given by f’(u) * u’(x).
• Example: Find the derivative of f(x) = ln(x^2 + 1).
  • Solution: Let u = x^2 + 1.
  • Using the chain rule, f’(x) = (1/u) * (2x).
  • Simplifying, f’(x) = (2x)/(x^2 + 1).

Example of finding derivatives of logarithmic functions

• Example 1: Find the derivative of f(x) = ln(3x^2 + 2).
  • Solution: Using the chain rule, f’(x) = (1/(3x^2 + 2)) * (6x).
  • Simplifying, f’(x) = 6x/(3x^2 + 2).
• Example 2: Find the derivative of f(x) = ln(2x^3 - 4x).
  • Solution: Using the chain rule, f’(x) = (1/(2x^3 - 4x)) * (6x^2 - 4).
  • Simplifying, f’(x) = (6x^2 - 4)/(2x^3 - 4x).

The Chain Rule for Logarithmic Functions

• The chain rule is a valuable technique for finding the derivative of composite functions, including logarithmic functions.
• The chain rule allows us to differentiate functions within functions.
• Example: Find the derivative of f(x) = ln(sin(x)).
  • Solution: Let u = sin(x).
  • Using the chain rule, f’(x) = (1/u) * cos(x).
  • Simplifying, f’(x) = cos(x)/sin(x) = cot(x).
• The chain rule extends the power of differentiation to more complex functions.

Application of Logarithmic Functions in Calculus

• Logarithmic functions find extensive applications in calculus, particularly in solving exponential growth and decay problems.
• They are used to model population growth, radioactive decay, compound interest, and many other phenomena.
• Logarithmic differentiation, which relies on logarithmic functions, is a powerful technique for finding derivatives when the function is expressed as the product, quotient, or power of other functions.
• The natural logarithm function, ln(x), is especially important in calculus as it helps simplify and solve various equations and integrals.

Summary of Properties of Logarithmic Functions

• Logarithmic functions have several essential properties, which are useful for various calculations and manipulations: 1. Logarithm of a product: log(ab) = log(a) + log(b) 2. Logarithm of a quotient: log(a/b) = log(a) - log(b) 3. Logarithm of a power: log(a^b) = b * log(a) 4. Logarithm of the base: log_b(b) = 1 5. Logarithm of 1: log_b(1) = 0 6. Change of base formula: log_a(x) = log_b(x)/log_b(a)
• These properties help simplify calculations, solve equations, and manipulate logarithmic functions efficiently.