Derivative of 1/x^2 is -2/x^3

Answer

-\frac{2}{x^{3}}

Derivative of 1/x^2

Rule Used: Quotient Rule
Steps: Identify a quotient in 1/ $x^2$ : use the quotient rule (u/v)' = (u'v − uv')/v². The minus sign belongs to the derivative of the denominator; swapping it is the usual error.
Differentiate numerator and denominator, then combine.
Final answer: - $2 \cdot x^{-3}$

The derivative of 1/x^2 is -2/x^3 for every real x except zero.

Instant step-by-step solution for 1/x^2

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Derivative of 1/x^2 = -2*x^(-3) Solve step-by-step for free: https://derivativecalculatorai.com/derivative-of-1-over-x-squared

Derivative

1/x^2 → -2*x^(-3)engine: -2*x^(-3)

Graph of the function and its derivative

The solid curve is f. The dashed curve is f′. Vertical gaps are asymptotes or jumps, not extra zeros.

Why the derivative has this form

Rewrite the reciprocal as a power before differentiating. 1/x^2 is x^(-2). The power rule multiplies by the old exponent and then subtracts 1, so -2 comes down and the exponent becomes -3. The 2 in the answer is that old exponent. It is not a chain-rule factor, and it is not optional.

The quotient rule on 1/x^2 reaches the same fraction. With numerator 1 and denominator x^2, the numerator of the derivative is 0·x^2 - 1·2x = -2x, and dividing by (x^2)^2 = x^4 cancels one x and leaves -2/x^3. If the power-rule answer and the quotient-rule answer disagree, a minus sign was dropped on one of the paths.

The original function is not defined at x = 0, and neither is the derivative: a negative power blows up there. For positive x the graph falls as x grows, so the slope is negative. For negative x, x^3 is negative, so -2/x^3 is positive and the left branch is rising toward the asymptote.

Power rule, then the quotient-rule check

Rewrite 1/x^2 as x^(-2). The exponent is -2, not 2.
Apply the power rule d/dx x^n = n x^(n-1) with n = -2.
Multiply by the exponent: -2 · x^(-2-1) = -2 x^(-3).
Write the negative power as a fraction: -2 x^(-3) = -2/x^3.
Check with the quotient rule. Let u = 1 and v = x^2, so u' = 0 and v' = 2x.
Numerator u'v - uv' = 0 - 2x = -2x. Denominator v^2 = x^4.
Simplify -2x/x^4 = -2/x^3. Both rules agree, for x ≠ 0.

Worked example

Slope of 1/x^2 at x = 2

The function value is 1/2^2 = 1/4.
The derivative formula gives -2/2^3 = -2/8 = -1/4.
A secant from 2 to 2.001: 1/(2.001)^2 is about 0.24975, and (0.24975 - 0.25)/0.001 is about -0.25.
The wrong answer -1/x^3 would give -1/8 at this point. The missing 2 is visible in the secant.

Answer: -1/4

Common mistakes

Dropping the 2 and answering -1/x^3

The exponent on 1/x^2 is -2, so the coefficient in front of the derivative is -2. -1/x^3 is the derivative of 1/x, moved to the wrong power. At x = 2 it is half the true slope.

Dropping the minus sign

2/x^3 is positive for positive x, but 1/x^2 is decreasing there. A positive derivative describes a reciprocal that climbs, and this one does not. The minus sign comes from the negative exponent.

Confusing 1/x^2 with 1/x

The derivative of 1/x is -1/x^2. That expression is the function on this page, not the answer. Differentiating again is what produces -2/x^3.

Derivative of sqrt(x) — 1/(2 sqrt(x)). Another power-rule rewrite, this time with exponent 1/2.
Quotient rule — The rule page lists this example and the others in the same family.
Power rule — The rule page lists this example and the others in the same family.
Derivative of 1/x — Same family of differentiation, different function.