Differentiating Logarithmic Functions
Differentiating logarithmic functions uses \(\dfrac{d}{dx}\ln x=\dfrac1x\) and, with the chain rule, \(\dfrac{d}{dx}\ln f(x)=\dfrac{f'(x)}{f(x)}\).
Part of the NSW Year 12 Mathematics Advanced course, in the Calculus area of study (Differential calculus focus area) of the 2024 syllabus. Work through practice questions with fully worked solutions and video lessons, or scroll down for the theory summary and worked examples.
Theory
The natural logarithm differentiates to \(\dfrac{1}{x}\). This Year 12 Mathematics Advanced topic (MAV-12-04) differentiates \(\ln x\) and \(\ln f(x)\) using the chain rule and log laws.
\(\dfrac{d}{dx}\ln x=\dfrac{1}{x}\). With the chain rule, \(\dfrac{d}{dx}\ln f(x)=\dfrac{f'(x)}{f(x)}\) — the derivative of the inside over the inside.
Log laws often simplify first: \(\ln(3x)=\ln3+\ln x\), so its derivative is just \(\dfrac{1}{x}\). The domain of \(\ln x\) is \(x>0\).
Method
- Simplify with log laws if possible.
- Identify the inside \(f(x)\) and its derivative.
- Write \(\dfrac{f'(x)}{f(x)}\).
| \(\ln(3x)\) | \(=\) | \(\ln3+\ln x\) |
| \(\dfrac{dy}{dx}\) | \(=\) | \(\dfrac{1}{x}\) |
| \(\dfrac{dy}{dx}\) | \(=\) | \(\dfrac{2x}{x^{2}+1}\) |
| \(\dfrac{dy}{dx}\) | \(=\) | \(\dfrac{5}{5x-2}\) |
| \(\dfrac{dy}{dx}\) | \(=\) | \(\dfrac{2}{x}\) |
| \(\text{at }x=1\) | \(=\) | \(2\) |
Common pitfalls
Frequently asked questions
What is the derivative of ln x?
It is 1 over x.
How do you differentiate ln of a function?
Use the chain rule: the derivative of ln f of x is f prime of x over f of x, the derivative of the inside divided by the inside.
What is the derivative of ln 3x?
Using log laws ln 3x equals ln 3 plus ln x, so the derivative is just 1 over x.
Why is the domain of ln x only positive numbers?
Because you can only take the logarithm of a positive number, so ln x is defined only for x greater than 0.