Lesson 7.4 – Derivative of ln x – AP Calculus AB – Mrs. Billerman
7.4 Derivative of ln x
Key Concepts & Processes
In this lesson, we will find the derivative of the natural logarithmic function y = ln x. We know that
it is differentiable because it is the inverse of the differentiable function 𝑦 = 𝑒𝑥.
𝑑
𝑑𝑥 (ln 𝑥) = 1
𝑥
With chain rule: 𝑑
𝑑𝑥 (ln 𝑢) = 1
𝑢
𝑑𝑢
𝑑𝑥 or 𝑑
𝑑𝑥 [ln(𝑔(𝑥)] =𝑔′(𝑥)
𝑔(𝑥)
Examples
EXAMPLE 1: Differentiate 𝑦 = ln(𝑥3+ 1).
EXAMPLE 2: Find 𝑑
𝑑𝑥 ln(sin 𝑥).
EXAMPLE 3: Differentiate 𝑓(𝑥)=√ln 𝑥.
