Derivative of Natural Log
The derivative of the natural logarithm.
Live Calculator
MembersVariables
Example
Slope of ln at x=1 is 1
How it works
The natural logarithm grows very slowly, and its rate of change at a positive value x is exactly 1/x. This happens because ln x is the inverse of e^x, so its slope must be the reciprocal of the exponential’s slope at the matching point; that reciprocal relationship gives 1/x. Since x must be positive, the formula applies only where ln x is defined as a real-valued function.
When to use it
Use this rule whenever you need the instantaneous rate of change of a logarithmic model, such as in growth/decay, information measures, or transformations involving logs. It is also essential when differentiating expressions like ln(3x+1) using the chain rule.
Worked example, step by step
Find the derivative of f(x) = ln x at x = 5. Use the formula: f'(x) = 1/x Substitute x = 5: f'(5) = 1/5 Convert to decimal if needed: f'(5) = 0.2 So the slope of the graph of ln x at x = 5 is 1/5, or 0.2.
Common mistakes to avoid
- Forgetting that x must be positive for ln x to be defined
- Writing the derivative as ln(1/x) instead of 1/x
- Using 1/(x+1) or another shifted form for the basic derivative of ln x
Did You Know?
The derivative of ln x is 1/x, quietly connecting logarithms to the area under a hyperbola.
Share this formula
More in Calculus
View allPower Rule (Derivative)
IntermediateDerivative of a power function.
Chain Rule
IntermediateDerivative of a composite function.
Product Rule
IntermediateDerivative of a product of two functions.
Quotient Rule
IntermediateDerivative of a quotient of two functions.
Derivative of Sine
IntermediateDerivative of the sine function.
Derivative of Cosine
IntermediateDerivative of the cosine function.