Derivative of 1/x

Answer

-\frac{1}{x^{2}}

Derivative of 1/x

Rule Used: Quotient Rule
Steps: Identify a quotient in 1/x: 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: - $x^{-2}$

d/dx 1/x = -x^(-2). The derivative exists for every real x except 0, where a denominator vanishes or a negative power blows up.

Instant step-by-step solution for 1/x

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

Derivative

1/x → -x^(-2)engine: -x^(-2)

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

1/x is the power x^(−1). Multiplying by the exponent −1 and subtracting 1 produces −x^(−2), which is −1/x^2. The quotient rule on 1/x gives the same fraction: numerator 0·x − 1·1, over x^2. The minus sign is the whole story of this derivative; without it the reciprocal would be described as increasing.

Apply the quotient rule to 1/x. The form to remember, already simplified, is -x^(-2).

The derivative exists for every real x except 0, where a denominator vanishes or a negative power blows up.

At x = 2, substitute into the simplified form -x^(-2). That number is the slope of 1/x there; a secant over a small step in x lands next to it.

Worked derivative

Write f(x) = 1/x.
Apply the quotient rule to 1/x. The form to remember, already simplified, is -x^(-2).
At x = 2, substitute into the simplified form -x^(-2). That number is the slope of 1/x there; a secant over a small step in x lands next to it.

Common mistakes

Answering 1/x^2 instead of −1/x^2

1/x^2 has the right denominator and the wrong sign. For positive x the reciprocal falls as x grows, so the slope is negative.

Dropping the minus on 1/x

Rewriting as a negative power makes the minus sign come from the exponent. A positive answer describes a reciprocal that grows, and these reciprocals do not.

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^2 — Same family of differentiation, different function.

Frequently asked questions

What is the derivative of 1/x?

-x^(-2). 1/x is the power x^(−1). Multiplying by the exponent −1 and subtracting 1 produces −x^(−2), which is −1/x^2. The quotient rule on 1/x gives the same fraction: numerator 0·x − 1·1, over x^2. The minus sign is the whole story of this derivative; without it the reciprocal would be described as increasing.