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derivative of ln x — Definition, Formula & Examples

The derivative of ln x is 1/x. This means that at any point on the curve y = ln x, the slope of the tangent line equals the reciprocal of the x-coordinate.

If f(x)=ln⁡xf(x) = \ln x for x>0x > 0, then f′(x)=1xf'(x) = \dfrac{1}{x}. More generally, by the chain rule, if u=g(x)u = g(x) is a differentiable function with g(x)>0g(x) > 0, then ddx[ln⁡(g(x))]=g′(x)g(x)\dfrac{d}{dx}[\ln(g(x))] = \dfrac{g'(x)}{g(x)}.

Key Formula

ddx[ln⁡x]=1x,x>0\frac{d}{dx}[\ln x] = \frac{1}{x}, \quad x > 0
Where:
  • xx = The input to the natural logarithm; must be positive
  • ln⁡\ln = The natural logarithm, i.e., logarithm base e

How It Works

To differentiate ln⁡x\ln x, you simply write 1x\frac{1}{x}. When the argument is more complex than just xx, apply the chain rule: differentiate the outer ln⁡\ln function to get one over the argument, then multiply by the derivative of the argument. For example, the derivative of ln⁡(3x2)\ln(3x^2) is 13x2⋅6x=2x\frac{1}{3x^2} \cdot 6x = \frac{2}{x}. This rule applies only when the argument is positive, since ln⁡\ln is defined only for positive inputs.

Worked Example

Problem: Find the derivative of f(x) = ln(5x³).
Identify the inner function: The argument of ln is u = 5x³, so you need the chain rule.
u=5x3u = 5x^3
Differentiate the inner function: Take the derivative of 5x³ with respect to x.
u′=15x2u' = 15x^2
Apply the chain rule: The derivative of ln(u) is 1/u times u'.
f′(x)=15x3⋅15x2=15x25x3f'(x) = \frac{1}{5x^3} \cdot 15x^2 = \frac{15x^2}{5x^3}
Simplify: Cancel common factors in the fraction.
f′(x)=3xf'(x) = \frac{3}{x}
Answer: f'(x) = 3/x

Another Example

Problem: Find the derivative of g(x) = x² · ln x.
Recognize the product: This is a product of two functions, so use the product rule: (uv)' = u'v + uv'.
u=x2,v=ln⁡xu = x^2, \quad v = \ln x
Differentiate each factor: Find u' and v' separately.
u′=2x,v′=1xu' = 2x, \quad v' = \frac{1}{x}
Apply the product rule: Multiply and add the two parts.
g′(x)=2x⋅ln⁡x+x2⋅1xg'(x) = 2x \cdot \ln x + x^2 \cdot \frac{1}{x}
Simplify: The second term simplifies to x.
g′(x)=2xln⁡x+xg'(x) = 2x\ln x + x
Answer: g'(x) = 2x ln x + x, which can be factored as x(2 ln x + 1)

Visualization

Why It Matters

The derivative of ln x appears constantly in AP Calculus AB/BC, from integration by parts to solving separable differential equations. In statistics and data science, the natural log transform is used to linearize exponential data, and understanding its derivative is essential for maximum likelihood estimation. Any field that models growth or decay — biology, economics, physics — relies on this rule routinely.

Common Mistakes

Mistake: Writing the derivative of ln x as x instead of 1/x.
Correction: The derivative of ln x is the reciprocal 1/x. You may be confusing this with the derivative of eˣ, which is eˣ. These are inverse functions, but their derivatives are not the same.
Mistake: Forgetting the chain rule when differentiating ln(g(x)).
Correction: If the argument is anything other than plain x, you must multiply 1/g(x) by g'(x). For instance, the derivative of ln(4x) is not 1/(4x) alone — it is (1/(4x)) · 4 = 1/x.

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