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Derivative of 1x\frac{1}{x}

The derivative of 1x\frac{1}{x} is −1x2\frac{-1}{x^{2}}.

f(x) =
ddx[1x]=−1x2\frac{d}{dx}\left[\frac{1}{x}\right] = \frac{-1}{x^{2}}

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Graph of f(x) = 1 / x and its derivative-3-2-1123-15-10-55slope -1f(x)f'(x)tangent at x = 1
Graph of f(x) = 1 / 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)=1xf(x) = \frac{1}{x} with respect to x. The rule that applies is the quotient rule.

    f(x)=1xf(x) = \frac{1}{x}
  2. 2

    Quotient rule: (uv)′=u′v−uv′v2\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^{2}}. Let u=1u = 1 and v=xv = x.

    u=1,v=xu = 1,\quad v = x
  3. 3

    Differentiate each piece:

    u′=0,v′=1u' = 0,\quad v' = 1
  4. 4

    Put them in the rule:

    f′(x)=(0)(x)−(1)(1)(x)2f'(x) = \frac{\left(0\right)\left(x\right) - \left(1\right)\left(1\right)}{\left(x\right)^{2}}
  5. 5

    Simplify:

    f′(x)=−1x2f'(x) = \frac{-1}{x^{2}}
  6. 6

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

    f′′(x)=−((−2)x)(x2)2f''(x) = \frac{-\left(\left(-2\right) x\right)}{\left(x^{2}\right)^{2}}
LaTeX source of the formulas on this page
\[ \frac{d}{dx}\left[\frac{1}{x}\right] = \frac{-1}{x^{2}} \]
\[ f(x) = \frac{1}{x} \]
\[ u = 1,\quad v = x \]
\[ u' = 0,\quad v' = 1 \]
\[ f'(x) = \frac{\left(0\right)\left(x\right) - \left(1\right)\left(1\right)}{\left(x\right)^{2}} \]
\[ f'(x) = \frac{-1}{x^{2}} \]
\[ f''(x) = \frac{-\left(\left(-2\right) x\right)}{\left(x^{2}\right)^{2}} \]

Values of f and f' for 1 / x

xf(x)f'(x)
-2-0.5-0.25
-1-1-1
0.52-4
11-1
20.5-0.25
30.33333333-0.11111111

Questions

What is the derivative of 1 / x?

The derivative of 1x\frac{1}{x} is −1x2\frac{-1}{x^{2}}.

What is the second derivative of 1 / x?

-(-2x) / (x2x^{2})^2.

Which rule do you use for 1 / x?

The quotient rule.

What is the slope of 1 / x at x = 1?

-1.

Related derivatives

    Derivative of 1 / x: -1 / x^2 (with steps) | WordGPT