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Powers, roots and polynomials

Derivative of 1/x²: Formula, Proof and Examples

The derivative of 1/x² is -2/x³. See the power rule and quotient rule proofs, the chain rule for 1/u², a graph and examples like 1/(3x - 2)².

Powers, roots and polynomials Updated 2 min read

At a glance

\frac{d}{dx}\,\frac{1}{x^2} = -\frac{2}{x^3}
Derivative
-\dfrac{2}{x^3} for x \ne 0
As a power
\dfrac{d}{dx}\,x^{-2} = -2x^{-3}
Chain rule form
\dfrac{d}{dx}\,\dfrac{1}{u^2} = -\dfrac{2u'}{u^3}
Second derivative
\dfrac{6}{x^4}
Integral version
\int \frac{1}{x^2}\,dx = -\frac{1}{x} + C
On this page
  1. Graph of 1/x² and its derivative
  2. Proof with the power rule
  3. Proof with the quotient rule
  4. Proof from the limit definition
  5. The chain rule version: derivative of 1/u²
  6. More worked examples
  7. How to remember it
  8. Common mistakes

The derivative of \dfrac{1}{x^2} is -\dfrac{2}{x^3}. Move the power to the top as x^{-2}, and the power rule finishes the job in one step.

\frac{d}{dx}\,\frac{1}{x^2} = -\frac{2}{x^3}, \qquad x \ne 0

Graph of 1/x² and its derivative

Graph of y = 1/x² and its derivative y = -2/x³
On the left, \frac{1}{x^2} rises toward the asymptote, so the derivative is positive. On the right it falls, so the derivative is negative.

The graph of \frac{1}{x^2} is symmetric about the y-axis, so its slopes at x and -x are opposites. That is why the derivative is an odd function.

Proof with the power rule

\frac{d}{dx}\,x^{-2} = -2x^{-2 - 1} = -2x^{-3} = -\frac{2}{x^3}

This is the power rule with n = -2. The only care needed is the subtraction: -2 - 1 = -3.

Proof with the quotient rule

With top 1 and bottom x^2, the quotient rule gives

\frac{0 \cdot x^2 - 1 \cdot 2x}{(x^2)^2} = \frac{-2x}{x^4} = -\frac{2}{x^3}

Same answer, more work. Rewriting as a power is the faster habit.

Proof from the limit definition

\frac{\frac{1}{(x + h)^2} - \frac{1}{x^2}}{h} = \frac{x^2 - (x + h)^2}{h\,x^2(x + h)^2} = \frac{-2x - h}{x^2(x + h)^2}

As h \to 0 this becomes \frac{-2x}{x^4} = -\frac{2}{x^3}.

The chain rule version: derivative of 1/u²

\frac{d}{dx}\,u^{-2} = -2u^{-3}\cdot u' = -\frac{2u'}{u^3}

Example 1. \frac{d}{dx}\,\frac{1}{(3x - 2)^2} = -\frac{2 \cdot 3}{(3x - 2)^3} = -\frac{6}{(3x - 2)^3}.

Example 2. \frac{d}{dx}\,\frac{1}{(x^2 + 4)^2} = -\frac{2 \cdot 2x}{(x^2 + 4)^3} = -\frac{4x}{(x^2 + 4)^3}.

Example 3. \frac{d}{dx}\,\frac{1}{\cos^2 x} = -\frac{2(-\sin x)}{\cos^3 x} = \frac{2\sin x}{\cos^3 x}. Since \frac{1}{\cos^2 x} = \sec^2 x, this matches the derivative of sec²x, 2\sec^2 x\tan x.

More worked examples

Example 4: a physics shape. Gravity and light intensity follow F = \frac{k}{r^2}. The rate of change with distance is \frac{dF}{dr} = -\frac{2k}{r^3}. Doubling the distance cuts the force to a quarter and the rate of change to an eighth.

Example 5: second derivative. \frac{d}{dx}\left(-2x^{-3}\right) = 6x^{-4} = \frac{6}{x^4}. It is positive everywhere, so both branches are concave up.

How to remember it

Think of \frac{1}{x^2} as a power that happens to sit in the denominator. Move it up as x^{-2}, apply the power rule like any other power, then move the result back down if you want a fraction. The same three steps work for \frac{1}{x^3}, \frac{1}{x^{10}} or \frac{1}{\sqrt{x}}. A quick sign check helps too: for positive x the curve is falling, so the derivative has to be negative there, and -\frac{2}{x^3} is.

Common mistakes

  1. Writing -2x^{-1}. Going down one from -2 lands at -3.
  2. Flipping the derivative of the bottom. \frac{1}{2x} is not the derivative of \frac{1}{x^2}.
  3. Losing the minus sign. The exponent -2 comes down as a negative coefficient.

Going backward, \int \frac{1}{x^2}\,dx = -\frac{1}{x} + C is on the integral of 1/x² page. Compare with the derivative of 1/x.

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Frequently asked questions

What is the derivative of 1/x²?
It is -\frac{2}{x^3}. Write \frac{1}{x^2} = x^{-2} and use the power rule: -2x^{-3}.

What is the derivative of 3/x²?
It is 3 \cdot \left(-\frac{2}{x^3}\right) = -\frac{6}{x^3}.

What is the derivative of 1/(x + 1)²?
By the chain rule, -2(x + 1)^{-3} \cdot 1 = -\frac{2}{(x + 1)^3}.

Is the derivative of 1/x² equal to 1/(2x)?
No. Taking the derivative of the bottom and flipping does not work. The correct answer is -\frac{2}{x^3}.

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Related derivatives

Derivative of 1/x \frac{d}{dx}\,\frac{1}{x} = -\frac{1}{x^2} The derivative of 1/x is -1/x². See the limit definition proof, the power rule with x⁻¹, the chain rule for 1/u and worked examples like 1/(2x + 5). Derivative of x² \frac{d}{dx}\,x^2 = 2x The derivative of x² is 2x. See the limit definition proof, the chain rule version for u², a graph, tangent lines and worked examples like (3x + 1)². Derivative of √x \frac{d}{dx}\,\sqrt{x} = \frac{1}{2\sqrt{x}} The derivative of √x is 1/(2√x). See the power rule and limit proofs, the chain rule for √u, why x = 0 fails, and examples like √(x² + 1).