Derivative of cot(x)

Answer

-\frac{\mathrm{sec}\left(x\right)^{2}}{\mathrm{tan}\left(x\right)^{2}}

Derivative of cot(x)

Rule Used: Trigonometric Rule
Steps: Recognize the trigonometric function $\cot(x)$ . These six derivatives are standard; a cofunction (cos, cot, csc) picks up a minus sign.
Apply the standard derivative: $\frac{d}{dx} \cot(x)$ = - $\sec(x)^2$ * $\tan(x)^{-2}$ .
Final answer: - $\sec(x)^2$ * $\tan(x)^{-2}$

d/dx cot(x) = -csc(x)^2. The derivative exists on the same open intervals as the function, and it blows up at the vertical asymptotes.

Instant step-by-step solution for cot(x)

Derivative TrigonometricIntermediate

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Derivative of cot(x) = -sec(x)^2*tan(x)^(-2) Solve step-by-step for free: https://derivativecalculatorai.com/derivative-of-cot-x

Derivative

cot(x) → -csc(x)^2engine: -sec(x)^2*tan(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

Cotangent is cos/sin, the cofunction of tangent. The quotient rule produces a numerator −sin^2 − cos^2 = −1, over sin^2, which is −csc^2(x). Every term in the numerator is negative, so there is no cancellation into a plus the way tangent's numerator became +1. Queries for the derivative of cot land on this sign.

Apply the trigonometric derivative to cot(x). The form to remember, already simplified, is -csc(x)^2.

The derivative exists on the same open intervals as the function, and it blows up at the vertical asymptotes.

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

Worked derivative

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

Common mistakes

Answering csc(x)^2

csc^2(x) without the minus is the derivative of −cot, or the pattern of sec^2 copied onto the wrong function. Cotangent decreases on (0, π).

Swapping cot(x) with its cofunction

Sine and cosine, tangent and cotangent, secant and cosecant are not interchangeable. The minus sign lives on the cofunction side.

Trigonometric derivatives — The rule page lists this example and the others in the same family.
Quotient rule — The rule page lists this example and the others in the same family.
Derivatives of trigonometric functions — Same family of differentiation, different function.
Derivative of sin(x) — Same family of differentiation, different function.
Derivative of cos(x) — Same family of differentiation, different function.
Derivative of tan(x) — Same family of differentiation, different function.
Derivative of sec(x) — Same family of differentiation, different function.
Derivative of csc(x) — Same family of differentiation, different function.

Frequently asked questions

What is the derivative of cot(x)?

-csc(x)^2. Cotangent is cos/sin, the cofunction of tangent. The quotient rule produces a numerator −sin^2 − cos^2 = −1, over sin^2, which is −csc^2(x). Every term in the numerator is negative, so there is no cancellation into a plus the way tangent's numerator became +1. Queries for the derivative of cot land on this sign.