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Exponential and log derivatives

Derivative of ln x: Formula, Proof and Examples

The derivative of ln x is 1/x. See why, with a proof from eʸ = x, the chain rule version for ln(u), a graph and worked examples like ln(2x) and ln(x²+1).

Exponential and log derivatives Updated 3 min read

At a glance

\frac{d}{dx}\,\ln x = \frac{1}{x}
Derivative
\dfrac{1}{x} for x > 0
Chain rule form
\dfrac{d}{dx}\ln u = \dfrac{u'}{u}
Absolute value form
\dfrac{d}{dx}\ln\lvert x\rvert = \dfrac{1}{x} for x \ne 0
Domain of ln x
x > 0
Integral version
\int \frac{1}{x}\,dx = \ln\lvert x\rvert + C
On this page
  1. Graph of ln x and its derivative
  2. Proof that the derivative of ln x is 1/x
  3. The chain rule version: derivative of ln(u)
  4. Worked examples
  5. Common mistakes

The derivative of \ln x is \dfrac{1}{x}. It is one of the most used results in calculus, and it fills a gap the power rule leaves: no power of x has \frac{1}{x} as its derivative, but the natural log does.

\frac{d}{dx}\,\ln x = \frac{1}{x}, \qquad x > 0

Graph of ln x and its derivative

Graph of y = ln x and its derivative y = 1/x
The slope of \ln x is large near x = 0 and shrinks as x grows. At x = 1 the slope is exactly 1, matching \frac{1}{1} = 1.

Since \frac{1}{x} is always positive for x > 0, the natural log is always increasing. And since \frac{1}{x} gets smaller as x grows, \ln x climbs more and more slowly, which is why it is concave down.

Proof that the derivative of ln x is 1/x

Let y = \ln x. By the definition of the natural log, this means

e^y = x

Differentiate both sides with respect to x, using implicit differentiation on the left:

e^y \cdot \frac{dy}{dx} = 1 \quad\Longrightarrow\quad \frac{dy}{dx} = \frac{1}{e^y} = \frac{1}{x}

The last step uses e^y = x again. The proof relies on the derivative of e^x being e^x, which you can read about on the derivative of eˣ page.

The chain rule version: derivative of ln(u)

Most logs you meet have something other than plain x inside. With the chain rule:

\frac{d}{dx}\,\ln u = \frac{1}{u}\cdot u' = \frac{u'}{u}

In words: the derivative of the inside, over the inside.

Function Inside u Derivative \frac{u'}{u}
\ln(2x) 2x \frac{2}{2x} = \frac{1}{x}
\ln(x^2 + 1) x^2 + 1 \frac{2x}{x^2 + 1}
\ln(\sin x) \sin x \frac{\cos x}{\sin x} = \cot x
\ln(e^x + 3) e^x + 3 \frac{e^x}{e^x + 3}
\ln(\ln x) \ln x \frac{1}{x\ln x}

Worked examples

Example 1: a product. Differentiate f(x) = x^3 \ln x with the product rule:

f'(x) = 3x^2 \ln x + x^3 \cdot \frac{1}{x} = 3x^2\ln x + x^2

Example 2: simplify first. Differentiate g(x) = \ln\!\left(\dfrac{x^4}{x + 1}\right). Log rules turn the quotient into a difference:

g(x) = 4\ln x - \ln(x + 1) \quad\Longrightarrow\quad g'(x) = \frac{4}{x} - \frac{1}{x + 1}

This is much faster than using the quotient rule inside the chain rule. Splitting logs before differentiating is the idea behind logarithmic differentiation.

Example 3: a tangent line. The tangent to y = \ln x at x = e has slope \frac{1}{e} and passes through (e, 1):

y - 1 = \frac{1}{e}(x - e) \quad\Longrightarrow\quad y = \frac{x}{e}

This line passes through the origin.

Common mistakes

  1. Writing \frac{1}{u} and stopping. \frac{d}{dx}\ln(x^2 + 1) is not \frac{1}{x^2+1}. Multiply by the inside derivative 2x.
  2. Treating \ln(x^2) and (\ln x)^2 as the same. \frac{d}{dx}\ln(x^2) = \frac{2}{x}, but \frac{d}{dx}(\ln x)^2 = \frac{2\ln x}{x}.
  3. Using \frac{1}{x} for other bases. \frac{d}{dx}\log_{10} x needs the extra factor \frac{1}{\ln 10}. See the derivative of log x.
  4. Forgetting the domain. \ln x only exists for x > 0. For negative inputs use \ln\lvert x\rvert, which has the same derivative \frac{1}{x}.

For more on logs and exponentials together, read derivatives of exponential and logarithmic functions, or see the natural logarithm function page.

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Frequently asked questions

What is the derivative of ln x?
The derivative of \ln x is \frac{1}{x}, for every x > 0.

What is the derivative of ln(2x)?
It is \frac{1}{x}. By the chain rule, \frac{d}{dx}\ln(2x) = \frac{2}{2x} = \frac{1}{x}. You can also see it from \ln(2x) = \ln 2 + \ln x, where \ln 2 is a constant.

What is the derivative of ln(x²)?
It is \frac{2x}{x^2} = \frac{2}{x}. You can also rewrite \ln(x^2) = 2\ln x first and get \frac{2}{x} directly.

Is the derivative of ln x the same as the derivative of log x?
Only for the natural log. For a base-10 log, \frac{d}{dx}\log_{10} x = \frac{1}{x\ln 10}. In general \frac{d}{dx}\log_b x = \frac{1}{x\ln b}.

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Related derivatives

Derivative of e^x \frac{d}{dx}\,e^x = e^x The derivative of eˣ is eˣ. See why with a limit proof, the chain rule form for e^u, a graph and worked examples like e^(3x) and x²eˣ. Derivative of log x \frac{d}{dx}\,\log_{10} x = \frac{1}{x\ln 10} The derivative of log₁₀ x is 1/(x ln 10), and of log_b x is 1/(x ln b). See the change of base proof, chain rule examples and a graph. Derivative of x ln x \frac{d}{dx}\,x\ln x = \ln x + 1 The derivative of x ln x is ln x + 1. See the product rule steps, the minimum at x = 1/e, the second derivative, a graph and similar examples.