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Reciprocal derivatives

Derivative of 1/x: Power Rule, Chain Rule and Domain

The derivative of 1/x is −1/x², not ln(x). Write the reciprocal as a negative power to see the sign and exponent clearly. The exclusion of zero remains part of the answer.

Direct answer

ddx(1x)=−1x2\frac{\mathrm{d}}{\mathrm{d}x}\left(\frac{1}{x}\right)=-\frac{1}{x^2}

The function and derivative are defined only for x ≠ 0. For 1/g(x), require g(x) ≠ 0 and differentiability.

Derivative of 1/x using the power rule

On either interval x<0 or x>0, write 1/x=x⁻¹. The rewrite does not give the function a value at zero.

Apply the power rule: multiply by −1 and lower the exponent to −2. Thus the derivative is −x⁻²=−1/x².

The quotient rule provides a second derivation: (1/x)′=(0·x−1·1)/x². It gives the same expression with the same excluded point.

For a reciprocal 1/g(x), use the chain rule: the outer derivative is −g(x)⁻² and the inner derivative is g′(x). Multiply them; do not replace g(x) by g′(x) in the denominator.

1
1x=x−1\frac{1}{x}=x^{-1}
2
ddxx−1=−x−2=−1x2\frac{\mathrm{d}}{\mathrm{d}x}x^{-1}=-x^{-2}=-\frac{1}{x^2}
3
ddx[1g(x)]=−g′(x)[g(x)]2\frac{\mathrm{d}}{\mathrm{d}x}\left[\frac{1}{g(x)}\right]=-\frac{g^{\prime}(x)}{[g(x)]^2}

Worked examples

ddx[1x]\frac{\mathrm{d}}{\mathrm{d}x}\left[\frac{1}{x}\right]
−1x2\frac{-1}{x^{2}}

Keep x ≠ 0; the slope is negative on both domain intervals.

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

The inner derivative is 2x; the denominator is positive for every real x.

ddx[13 x+2]\frac{\mathrm{d}}{\mathrm{d}x}\left[\frac{1}{3\,x + 2}\right]
−3(3 x+2)2\frac{-3}{\left(3\,x + 2\right)^{2}}

The inner derivative is 3; exclude x=−2/3.

Common mistakes

  • • Using the integral ln|x| instead of the derivative.
  • • Dropping the minus sign or failing to square the denominator.
  • • Allowing x=0 because a simplified expression looks familiar.
  • • Forgetting g′(x) when the denominator is composite.

Try the worked examples in the calculator

Each link opens the matching calculator with the expression, variable and required conditions already filled in.

求导:1x\frac{1}{x}求导:1x2+1\frac{1}{x^{2} + 1}求导:13 x+2\frac{1}{3\,x + 2}
Open the calculator without a preset: 求导

References

Frequently asked questions

Is the derivative of 1/x positive for negative x?+

No. x² is positive whenever x ≠ 0, so −1/x² is negative on both sides of zero.

Is 1/x differentiable at zero?+

No. The original function is undefined there, so it has no derivative at zero.

What is the difference between differentiating and integrating 1/x?+

Differentiating gives −1/x². Integrating gives ln|x|+C on an interval excluding zero.

Conclusion

For the derivative of 1/x, use exponent −1, retain the negative sign and exclude zero. For a composite denominator, multiply by its derivative before simplifying.

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