Derivative of 1/x
1/x hides a power rule problem in fraction clothing — rewrite it as x⁻¹ and the answer is two mechanical steps away.
Step-by-step solution
1/x = x−1. A reciprocal is a negative exponent.
d/dx[x−1] = (−1)·x−2 — coefficient down, exponent reduced from −1 to −2.
f′(x) = −1/x2.
Why it works
The minus sign is the whole personality of this function: 1/x is decreasing on both of its branches, and its derivative is strictly negative on each branch. Meanwhile the x² in the denominator says the curve flattens rapidly as you move away from the origin and steepens violently as you approach it — matching the hyperbola's shape exactly.
Note what the derivative does not say: it does not connect the two branches. Both branches slope downward, but the function still jumps from −∞ to +∞ across x = 0, where neither the function nor its derivative exists. "Negative derivative everywhere it's defined" and "decreasing over its whole domain" are subtly different claims, and 1/x is the classic counterexample that keeps that distinction honest.
Domain and the limit definition
Both 1/x and its derivative −1/x² are defined for x ≠ 0. The derivative is negative on each interval (−∞, 0) and (0, ∞). There is no function value or tangent line at zero, and the two branches must be considered separately.
For a fixed nonzero x, the difference quotient is [1/(x + h) − 1/x]/h = −1/[x(x + h)]. Letting h approach zero gives −1/x². At x = 2, the point is (2, 1/2) and the tangent line is y = 1/2 − (x − 2)/4.
An antiderivative on either branch is ln|x| + C. Writing ln(x) alone covers only the positive branch. Differentiation and integration are different operations; neither gives a value at the excluded input zero.
Common mistakes
- Dropping the minus sign — the most frequent error, since the power rule's n = −1 supplies it automatically but it's easy to lose in the rewrite.
- Answering ln|x|: that's an antiderivative of 1/x, not the derivative. The two directions get swapped constantly.
- Writing −1/x after reducing the exponent incorrectly (−1 − 1 = −2, not 0 or −1).
Practice problems
Differentiate 3/x
Answer: −3/x².
Differentiate 1/x²
Answer: −2/x³ — same method with n = −2.
Find the slope of 1/x at x = 2
Answer: −1/4.