Derivative of ln x
Derivative of the natural logarithm.
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Example
The slope of ln x at x = 2 is 1/2.
How it works
The natural logarithm changes more and more slowly as x gets larger, and its slope at any positive x is exactly 1/x. This comes from the inverse relationship between ln x and e^x: since e^y = x implies y = ln x, differentiating this identity gives dy/dx = 1/x. So the derivative is positive for all x > 0, but it gets smaller as x increases.
When to use it
Use this rule whenever you need the slope or rate of change of ln x, such as in calculus problems, optimization, or modeling growth and scaling effects. It is also a key building block for differentiating expressions like ln(g(x)) using the chain rule.
Worked example, step by step
Find the derivative of f(x) = ln x at x = 5. Step 1: Use the rule d/dx(ln x) = 1/x. Step 2: Substitute x = 5 into the derivative: f'(5) = 1/5. Step 3: Write the decimal if needed: f'(5) = 0.2. Final answer: The slope of ln x at x = 5 is 1/5.
Common mistakes to avoid
- Differentiating ln x as 1/ln x instead of 1/x
- Forgetting that ln x is only defined for x > 0 in real calculus
- Applying the rule to ln(g(x)) without using the chain rule
Did You Know?
The derivative of ln x is 1/x, which is why the natural log appears whenever you integrate 1/x.
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