Derivatives of Logarithms

Business Calculus · Axiom Academy

LESSON Derivatives of Logarithmic Functions The inverse relationship between logs and exponentials The natural logarithm (x) is the inverse of e^x . Its derivative is surprisingly simple: The derivative of ln(x) is simply 1/x Notice that the slope of (x) decreases as x increases. At x = 1 , the slope is 1. At x = 2 , the slope is 2 . (x) is only defined for x > 0 , so its derivative x is also only valid for x > 0 . What about logarithms with bases other than e ? The natural log of the base appears in the denominator Since (e) = 1 , the natural log has the simplest derivative. That's why we typically convert other logs to natural logs for calculus! When the argument of the log is a function, we need the chain rule: Divide by the inside, multiply by its derivative Example 1: Find the derivative of f(x) = (3x) Example 2: Find the derivative of g(x) = (x^2 + 1) Sometimes it's easier to simplify using log properties before differentiating: Example: Find the derivative of f(x) = (x^3) Method 1: Use log properties first Method 2: Use chain rule directly Both methods give the same answer! Click each problem to reveal the solution: Business Application: Growth Rate If P(t) represents population or revenue over time, the relative growth rate is: This is exactly the derivative of (P(t)) ! Find the derivative of f(x) = (2x + 5) You've mastered logarithmic derivatives! Combined with exponential derivatives, you now have all the tools for business calculus applications!

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