Natural logarithms

Derivative of ln(x): Formula, Proof, and Examples

The natural logarithm is the inverse of the exponential function. Its derivative is reciprocal because equal multiplicative changes in x become equal additive changes in ln(x).

Direct answer

ddx[ln⁡x]=1x\frac{\mathrm{d}}{\mathrm{d}x}\left[\ln x\right] = \frac{1}{x}

For real-valued ln(x), x > 0.

Proof by inverse differentiation

Let y = ln(x). The inverse relationship is x = exp(y). Differentiate both sides with respect to x.

The chain rule gives 1 = exp(y) · dy/dx. Since exp(y) = x, divide by x to obtain dy/dx = 1/x.

For ln|x|, the same derivative holds on any real interval that does not cross zero. This is why the antiderivative of 1/x is ln|x| + C, not merely ln(x) + C.

1
y = ln(x) ⇔ x = exp(y)
2
1 = exp(y) · dy/dx
3
dydx=1x\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{1}{x}

Worked examples

ddx[ln⁡(5 x)]\frac{\mathrm{d}}{\mathrm{d}x}\left[\ln\left(5\,x\right)\right]
1x\frac{1}{x}

The constant 5 cancels after the chain rule.

ddx[ln⁡(x2+1)]\frac{\mathrm{d}}{\mathrm{d}x}\left[\ln\left(x^{2} + 1\right)\right]
2 xx2+1\frac{2\,x}{x^{2} + 1}

Differentiate the inner expression and divide by it.

ddx[(ln⁡x)2]\frac{\mathrm{d}}{\mathrm{d}x}\left[\left(\ln x\right)^{2}\right]
2 ln⁡xx\frac{2\,\ln x}{x}

Apply the outer power rule, then the logarithm rule.

Common mistakes

  • • Writing ln(x) on x < 0 in real calculus; ln|x| is a different function with a larger domain.
  • • Assuming every calculator button labeled log has base 10. In CalculusMate, ln and log both parse as the natural logarithm, base e.
  • • Forgetting the inner derivative in ln(g(x)): the result is g′(x)/g(x).

Try the worked examples in the calculator

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Derivatives:ln⁡(x2+1)\ln\left(x^{2} + 1\right)
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