Derivatives of Logarithmic and General Exponential Functions

Basic Differentiation Rule: The derivative of the natural logarithmic function is given by: \frac{d}{dx}[\ln(x)] = \frac{1}{x}, \, x > 0
Chain Rule Version: If $u$ is a differentiable function of $x$ , then: \frac{d}{dx}[\ln(u)] = \frac{1}{u} \frac{du}{dx}, \, u > 0

Composite Functions: For $f(x) = \ln(2x^2 + 4)$ , the derivative is: $f'(x) = \frac{1}{2x^2 + 4}(4x) = \frac{2x}{x^2 + 2}$
Product Rule: For $f(x) = x \ln(x)$ , the derivative is: $f'(x) = x\left(\frac{1}{x}\right) + (\ln(x))(1) = 1 + \ln(x)$
Quotient Rule: For $f(x) = \frac{\ln(x)}{x}$ , the derivative is: $f'(x) = \frac{x\left(\frac{1}{x}\right) - (\ln(x))(1)}{x^2} = \frac{1 - \ln(x)}{x^2}$

Definition: For a positive number $a$ where $a \neq 1$ , the general logarithmic function is defined by: $\log_a(x) = y \iff a^y = x$
Change-of-Base Formula: Logarithms to bases other than $e$ can be evaluated by converting to natural logarithms: $\log_a(x) = \frac{\ln(x)}{\ln(a)}$
Differentiation: Derivatives of exponential or logarithmic functions with bases other than $e$ can be found by either converting to base $e$ or using specific differentiation rules for base $a$ .

Scenario: Radioactive carbon isotopes have a half-life of $5715$ years. A sample contains $1$ gram of the isotope.
Rate of Change: When $t = 10,000$ years, the rate at which the amount of isotope is changing is calculated to be approximately $-0.000036$ .
Conclusion: The amount of isotopes in the object decreases at a rate of $0.000036\,g/\text{year}$ at the $10,000$ -year mark.