AP Calculus AB and BC

Derivative of 1/x^2: Answer, Power Rule, Mistakes

The derivative of 1 over x squared with respect to x is negative 2 over x cubed. Rewrite the expression as x to the negative two and apply the power rule, which brings the negative exponent down in front and lowers it to negative three.

ddx[1x2]=−2x3\frac{d}{dx}\left[\frac{1}{x^{2}}\right] = -\frac{2}{x^{3}}

How to differentiate 1/x^2

Move the power upstairs first. There is no quotient rule needed once the expression is written as a single power.

1x2=x−2\frac{1}{x^{2}} = x^{-2}

Now the power rule applies with n=−2n = -2: multiply by −2-2 and lower the exponent to −3-3.

ddxx−2=−2x−3=−2x3\frac{d}{dx}x^{-2} = -2x^{-3} = -\frac{2}{x^{3}}

Reading the sign

For x>0x > 0 the derivative is negative, which is right: the function is falling toward zero. For x<0x < 0 the cube in the denominator is negative, so the derivative is positive and the function is rising.

The function itself is always positive and even, so its derivative must be odd, which −2x3-\frac{2}{x^{3}} is.

Where the derivative of 1/x^2 shows up on the AP exam

Topic 2.5 on both exams. Inverse square relationships appear throughout related rates and physics contexts, and this is that derivative wearing different letters.

Neither the function nor its derivative exists at x=0x = 0, where there is a vertical asymptote.

Common mistakes with the derivative of 1/x^2

  • Answering 12x\frac{1}{2x} or −12x-\frac{1}{2x}, which come from mishandling the reciprocal rather than rewriting as a power.
  • Answering −2x−3-2x^{-3} but then writing it as −2x−3-\frac{2}{x^{-3}}. A negative exponent moves the factor to the denominator with a POSITIVE exponent.
  • Losing the minus sign, which would claim the function increases for positive xx.
  • Confusing it with the antiderivative −1x+C-\frac{1}{x} + C, which is a different question about the same expression.

Check yourself, not just the answer

Type derivatives and get graded on mathematical equivalence, with rule-level hints when you miss, in the Derivative Practice Checker.

Frequently asked questions

What is the derivative of 1/x^2?

It is −2x3-\frac{2}{x^{3}}.

Why rewrite it as x^-2 first?

Because the power rule needs a single power. Written as a fraction it looks like a quotient rule problem, which is slower and easier to get wrong.

What is the integral of 1/x^2?

It is −1x+C-\frac{1}{x} + C. Do not confuse the derivative with the antiderivative; both carry a minus sign but they are different expressions.

Is the derivative always negative?

No. It is negative for x>0x > 0 and positive for x<0x < 0, because the cube in the denominator changes sign.