Derivative of csc(x)

Answer

-\frac{\mathrm{cos}\left(x\right)}{\mathrm{sin}\left(x\right)^{2}}

Derivative of csc(x)

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

d/dx csc(x) = -cot(x)*csc(x). 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 csc(x)

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Derivative of csc(x) = -cos(x)*sin(x)^(-2) Solve step-by-step for free: https://derivativecalculatorai.com/derivative-of-csc-x

Derivative

csc(x) → -cot(x)*csc(x)engine: -cos(x)*sin(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

Cosecant is 1/sin(x), the reciprocal partner of secant. The same reciprocal differentiation now multiplies by the derivative of sine, which is +cos, so one minus remains: −cos/sin^2 = −cot(x) csc(x). The minus is the difference between csc and sec. Secant lost its minus because cosine's derivative already had one; sine's derivative does not.

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

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 -cot(x)*csc(x). That number is the slope of csc(x) there; a secant over a small step in x lands next to it.

Worked derivative

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

Common mistakes

Answering cot(x)*csc(x)

Dropping the minus copies the shape of sec·tan. Cosecant is the cofunction, and cofunctions pick up the minus.

Swapping csc(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 cot(x) — Same family of differentiation, different function.

Frequently asked questions

What is the derivative of csc(x)?

-cot(x)*csc(x). Cosecant is 1/sin(x), the reciprocal partner of secant. The same reciprocal differentiation now multiplies by the derivative of sine, which is +cos, so one minus remains: −cos/sin^2 = −cot(x) csc(x). The minus is the difference between csc and sec. Secant lost its minus because cosine's derivative already had one; sine's derivative does not.