Derivative of sec(x)

Answer

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

Derivative of sec(x)

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

d/dx sec(x) = sec(x)*tan(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 sec(x)

Derivative TrigonometricIntermediate

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

Derivative

sec(x) → sec(x)*tan(x)engine: cos(x)^(-2)*sin(x)

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

Secant is 1/cos(x). Differentiating the reciprocal brings −1/cos^2 times the derivative of cosine, which is −sin, so the two minuses cancel and you get sin/cos^2. That splits as (1/cos)·(sin/cos) = sec(x) tan(x). The product, not a lone tangent, is the derivative.

Apply the trigonometric derivative to sec(x). The form to remember, already simplified, is sec(x)*tan(x).

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

At x = 0.5, substitute into the simplified form sec(x)*tan(x). That number is the slope of sec(x) there; a secant over a small step in x lands next to it.

Worked derivative

Write f(x) = sec(x).
Apply the trigonometric derivative to sec(x). The form to remember, already simplified, is sec(x)*tan(x).
At x = 0.5, substitute into the simplified form sec(x)*tan(x). That number is the slope of sec(x) there; a secant over a small step in x lands next to it.

Common mistakes

Answering tan(x)

tan(x) alone drops the secant factor. Both pieces come from sin/cos^2, and neither is optional.

Swapping sec(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 csc(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 sec(x)?

sec(x)*tan(x). Secant is 1/cos(x). Differentiating the reciprocal brings −1/cos^2 times the derivative of cosine, which is −sin, so the two minuses cancel and you get sin/cos^2. That splits as (1/cos)·(sin/cos) = sec(x) tan(x). The product, not a lone tangent, is the derivative.