9.2 Natural logarithm and its derivatives

The natural logarithm function is a powerful tool in calculus, with unique properties that make it essential for solving complex problems. It's the inverse of the exponential function and has a special relationship with Euler's number, e.

Understanding the derivative of the natural logarithm is crucial. Its simple form, 1/x, leads to elegant solutions in optimization and analysis. This function's properties and derivatives are key to tackling advanced calculus problems.

The Natural Logarithm Function

Natural logarithm function properties

• Denoted as $\ln(x)$, logarithm with base $e$ (Euler's number, approximately 2.71828)
• Defined for all positive real numbers, domain is $x > 0$
• $\ln(1) = 0$ since $e^0 = 1$
• $\ln(e) = 1$ since $e^1 = e$
• $\ln(e^x) = x$ for all real numbers $x$, logarithm and exponential cancel each other
• $e^{\ln(x)} = x$ for all $x > 0$, exponential and logarithm cancel each other
• $\ln(xy) = \ln(x) + \ln(y)$ for all $x, y > 0$, logarithm of a product is the sum of logarithms
• $\ln(\frac{x}{y}) = \ln(x) - \ln(y)$ for all $x, y > 0$, logarithm of a quotient is the difference of logarithms
• $\ln(x^n) = n \ln(x)$ for all $x > 0$ and real numbers $n$, logarithm of a power is the product of the exponent and logarithm

Derivatives of the Natural Logarithm Function

Derivative of natural logarithm

• $\frac{d}{dx} \ln(x) = \frac{1}{x}$ for all $x > 0$
• Proof using limit definition of derivative:
  1. Consider $\lim_{h \to 0} \frac{\ln(x+h) - \ln(x)}{h}$
  2. Rewrite numerator using logarithm properties: $\ln(x+h) - \ln(x) = \ln(\frac{x+h}{x})$
  3. Simplify limit: $\lim_{h \to 0} \frac{\ln(\frac{x+h}{x})}{h} = \lim_{h \to 0} \frac{\ln(1+\frac{h}{x})}{h}$
  4. Substitute $u = \frac{h}{x}$, then $h = xu$ and as $h \to 0$, $u \to 0$: $\lim_{u \to 0} \frac{\ln(1+u)}{xu} = \frac{1}{x} \lim_{u \to 0} \frac{\ln(1+u)}{u}$
  5. $\lim_{u \to 0} \frac{\ln(1+u)}{u} = 1$ (well-known limit), so $\frac{d}{dx} \ln(x) = \frac{1}{x}$

Applications of logarithmic derivatives

• Differentiate functions involving natural logarithms:
  • $\frac{d}{dx} \ln(2x) = \frac{1}{2x} \cdot \frac{d}{dx}(2x) = \frac{1}{2x} \cdot 2 = \frac{1}{x}$
  • $\frac{d}{dx} \ln(x^2+1) = \frac{1}{x^2+1} \cdot \frac{d}{dx}(x^2+1) = \frac{1}{x^2+1} \cdot 2x = \frac{2x}{x^2+1}$
• Solve optimization problems involving natural logarithms
• Analyze behavior of functions involving natural logarithms using derivatives

Natural logarithm vs exponential functions

• Natural logarithm and exponential functions are inverses of each other:
  • If $y = \ln(x)$, then $x = e^y$
  • If $y = e^x$, then $x = \ln(y)$
• Derivatives of exponential functions:
  • If $f(x) = e^x$, then $f'(x) = e^x$
  • If $f(x) = e^{g(x)}$, then $f'(x) = e^{g(x)} \cdot g'(x)$ (chain rule)
• Derivatives of natural logarithm and exponential functions are reciprocals:
  • $\frac{d}{dx} e^x = e^x$ and $\frac{d}{dx} \ln(x) = \frac{1}{x}$