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Derivative of ln⁡(x)\ln\left(x\right)

The derivative of ln⁡(x)\ln\left(x\right) is 1x\frac{1}{x}.

f(x) =
ddx[ln⁡(x)]=1x\frac{d}{dx}\left[\ln\left(x\right)\right] = \frac{1}{x}

Type any function of x; the derivative is computed symbolically in your browser.

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Graph of f(x) = ln(x) and its derivative-3-2-1123-4-3-2-1123slope 1f(x)f'(x)tangent at x = 1
Graph of f(x) = ln(x) and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = 1.

How to work it out

  1. 1

    We differentiate f(x)=ln⁡(x)f(x) = \ln\left(x\right) with respect to x. The rule that applies is the standard derivative.

    f(x)=ln⁡(x)f(x) = \ln\left(x\right)
  2. 2

    This is a standard derivative from the table.

  3. 3

    Simplify:

    f′(x)=1xf'(x) = \frac{1}{x}
  4. 4

    Differentiating once more gives the second derivative, which tells you about curvature:

    f′′(x)=−1x2f''(x) = \frac{-1}{x^{2}}
LaTeX source of the formulas on this page
\[ \frac{d}{dx}\left[\ln\left(x\right)\right] = \frac{1}{x} \]
\[ f(x) = \ln\left(x\right) \]
\[ f'(x) = \frac{1}{x} \]
\[ f''(x) = \frac{-1}{x^{2}} \]

Values of f and f' for ln(x)

xf(x)f'(x)
0.5-0.693147182
101
20.693147180.5
31.09861230.33333333

Questions

What is the derivative of ln(x)?

The derivative of ln⁡(x)\ln\left(x\right) is 1x\frac{1}{x}.

What is the second derivative of ln(x)?

-1 / x2x^{2}.

Which rule do you use for ln(x)?

The standard derivative.

What is the slope of ln(x) at x = 1?

1.

Related derivatives

    Derivative of ln(x): 1 / x (with steps) | WordGPT