Derivatives - ln(x) as inverse of e^x

  1. Introduction to ln(x) as inverse of e^x
  • The natural logarithm function ln(x) is defined as the inverse of the exponential function e^x.
  • It is commonly used in various fields including calculus and mathematical modeling.
  1. Definition of the natural logarithm ln(x)
  • The natural logarithm ln(x) represents the power to which the number e (approximately 2.71828) must be raised to obtain the value x.
  • It is denoted as ln(x) or loge(x).
  • For example, ln(1) = 0, ln(e) = 1, ln(2) ≈ 0.6931, etc.
  1. Properties of ln(x)
  • Property 1: ln(1) = 0
  • Property 2: ln(e) = 1
  • Property 3: ln(x * y) = ln(x) + ln(y)
  • Property 4: ln(x / y) = ln(x) - ln(y)
  • Property 5: ln(x^n) = n * ln(x)
  1. Derivative of ln(x) using the definition
  • To find the derivative of ln(x), we can use the definition of the derivative.
  • Let f(x) = ln(x), then f’(x) = lim(h -> 0) [(ln(x + h) - ln(x)) / h]
  • Simplifying the expression, f’(x) = 1 / x
  1. Example 1: Finding the derivative of ln(x)
  • Let’s find the derivative of f(x) = ln(x).
  • f’(x) = 1 / x
  • For example, if we want to find the derivative of ln(2), we substitute x = 2 in the derivative expression: f’(2) = 1 / 2.
  1. Derivative of ln(x) using logarithmic differentiation
  • Another method to find the derivative of ln(x) is by using logarithmic differentiation.
  • Let’s consider a function y = ln(f(x)), where f(x) is some function of x.
  • Taking the natural logarithm of both sides, we get ln(y) = ln(ln(f(x))).
  • Applying the derivative on both sides, we get: 1 / y * y’ = 1 / f(x) * f’(x)
  • Simplifying the expression, y’ = f’(x) / f(x)
  1. Derivative of ln(x) using logarithmic differentiation (contd.)
  • Applying logarithmic differentiation to ln(x), we have y = ln(x), where f(x) = x.
  • Taking the natural logarithm of both sides, we get ln(y) = ln(ln(x)).
  • Applying the derivative on both sides, we get: 1 / y * y’ = 1 / x
  • Simplifying the expression, y’ = 1 / (x * y), where y = ln(x)
  1. Example 2: Finding the derivative of ln(x)
  • Let’s find the derivative of f(x) = ln(x).
  • Using logarithmic differentiation, y = ln(x), where f(x) = x.
  • Differentiating both sides, we get: y’ = 1 / (x * y)
  • Substituting the value of y = ln(x), we have: y’ = 1 / (x * ln(x))
  1. Derivative of ln(x) using the chain rule
  • The chain rule can also be used to find the derivative of ln(x).
  • Let’s consider a function y = ln(f(x)), where f(x) is some function of x.
  • Using the chain rule, we have: y’ = f’(x) / f(x)
  1. Derivative of ln(x) using the chain rule (contd.)
  • Applying the chain rule to ln(x), we have y = ln(x), where f(x) = x.
  • Differentiating both sides, we get: y’ = f’(x) / f(x)
  • Substituting the value of f(x) = x and f’(x) = 1, we have: y’ = 1 / x
  1. Properties of ln(x) (continued)
  • Property 6: ln(e^x) = x
  • Property 7: ln(x^a) = a * ln(x)
  • Property 8: ln(1/x) = -ln(x)
  1. Derivative of ln(x) using the chain rule (continued)
  • The chain rule can be applied to more complex functions involving ln(x).
  • Example: Find the derivative of f(x) = ln(2x + 1).
    • Let g(x) = 2x + 1, then f(x) = ln(g(x)).
    • Using the chain rule, f’(x) = g’(x) / g(x) = 2 / (2x + 1).
  1. Derivative of ln(x^n)
  • We can find the derivative of ln(x^n) using the chain rule.
  • Example: Find the derivative of f(x) = ln(x^3).
    • Using the chain rule, we have f’(x) = (x^3)’ / (x^3) = 3x^2 / x^3 = 3 / x.
  1. Applications of ln(x) derivatives
  • The derivative of ln(x) has applications in various areas, including population growth, compound interest, and exponential decay.
  • It is often used to model growth and decay in real-world scenarios.
  • Example: Modeling population growth using the function P(t) = P0 * e^(kt), where P0 is the initial population, t is time, and k is the growth rate.
  1. Example 3: Modeling population growth
  • Let’s consider a scenario where the initial population P0 is 1000 and the growth rate k is 0.05. Find the derivative of ln(P(t)) with respect to t.
    • P(t) = 1000 * e^(0.05t)
    • Taking the natural logarithm of both sides, we have ln(P(t)) = ln(1000) + 0.05t
    • The derivative of ln(P(t)) with respect to t is equal to the growth rate constant k. Therefore, the derivative is 0.05.
  1. Example 4: Compound interest
  • The derivative of ln(x) can be used to calculate the continuous compound interest.
  • Let’s consider a scenario where the initial investment A0 is $1000 and the annual interest rate r is 5%. Find the derivative of ln(A(t)) with respect to t.
    • A(t) = A0 * e^(rt)
    • Taking the natural logarithm of both sides, we have ln(A(t)) = ln(A0) + rt
    • The derivative of ln(A(t)) with respect to t is equal to the interest rate r. Therefore, the derivative is 0.05.
  1. Example 5: Exponential decay
  • The derivative of ln(x) can also be used to model exponential decay.
  • Let’s consider a scenario where the initial quantity Q0 is 1000 and the decay rate k is 0.05. Find the derivative of ln(Q(t)) with respect to t.
    • Q(t) = Q0 * e^(-kt)
    • Taking the natural logarithm of both sides, we have ln(Q(t)) = ln(Q0) - kt
    • The derivative of ln(Q(t)) with respect to t is equal to the negative decay rate constant k. Therefore, the derivative is -0.05.
  1. Summary of derivatives of ln(x)
  • The derivative of ln(x) can be calculated using different methods, such as the definition, logarithmic differentiation, and the chain rule.
  • It has several key properties, including ln(1) = 0, ln(e) = 1, and ln(x * y) = ln(x) + ln(y).
  • The derivative is given by f’(x) = 1 / x, where f(x) = ln(x).
  1. Practice problems
  • Find the derivative of f(x) = ln(3x + 2).
  • Find the derivative of g(x) = ln(5e^x).
  • Find the derivative of h(x) = ln(sqrt(x)).
  • Find the derivative of m(x) = ln(x^4).
  • Find the derivative of n(x) = ln(2 / x).
  1. Conclusion
  • The natural logarithm function ln(x) is the inverse of the exponential function e^x.
  • Its derivative can be found using various methods, such as the definition, logarithmic differentiation, and the chain rule.
  • Understanding the properties and applications of ln(x) derivatives is important in calculus and mathematical modeling.