Derivative of tan(x) is sec(x)^2

Answer

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

Derivative of tan(x)

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

The derivative of tan(x) is sec(x)^2 wherever cos(x) is not zero.

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

Derivative TrigonometricIntermediate

Share this result

Share on X

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

Derivative

tan(x) → sec(x)^2engine: sec(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

That is the same function as 1/cos(x)^2, and the Pythagorean identity also writes it as 1 + tan(x)^2. The three forms agree on every interval where tangent itself is defined. They are not three different derivatives.

Tangent is the quotient sin(x)/cos(x). The quotient rule is what produces the square: the denominator of the rule is already cos(x)^2, and the numerator collapses to 1 because cos(x)^2 + sin(x)^2 = 1. A calculator that only prints "trigonometric rule" is skipping that algebra. The algebra is the reason the square has to be there.

Cosine is zero at the odd multiples of pi/2, and tangent has a vertical asymptote at each of those points. The derivative is undefined in the same places. Between the asymptotes, sec(x)^2 is always positive, so tangent is increasing on each interval where it exists. Near x = 0 the value is about 1, which matches the slope you see on the graph.

Quotient-rule derivation of sec(x)^2

Write tan(x) as sin(x)/cos(x). Call the numerator u and the denominator v.
Differentiate the pieces you already know: u' = cos(x) and v' = -sin(x).
Apply the quotient rule (u/v)' = (u'v - uv') / v^2.
Substitute: numerator = cos(x)·cos(x) - sin(x)·(-sin(x)) = cos(x)^2 + sin(x)^2.
Replace the numerator with 1, using cos(x)^2 + sin(x)^2 = 1. The denominator stays cos(x)^2.
Conclude d/dx tan(x) = 1/cos(x)^2. By the definition of secant, that is sec(x)^2.
Optional check of the other form: 1 + tan(x)^2 = 1 + sin(x)^2/cos(x)^2 = (cos(x)^2 + sin(x)^2)/cos(x)^2 = sec(x)^2.

Worked example

Slope of tan(x) at x = pi/4

sec(pi/4) = 1/cos(pi/4) = 1/(√2/2) = √2.
Square it: sec(pi/4)^2 = 2. The slope of the tangent line is 2, not √2.
tan(pi/4) = 1, so the tangent line is y - 1 = 2(x - pi/4).
A secant from pi/4 to pi/4 + 0.001 has a slope near 2. It does not approach 1, which is what you would get from sec(x) without the square.

Answer: 2

Common mistakes

Answering sec(x) instead of sec(x)^2

sec(x) is the reciprocal of cosine. The quotient rule leaves cos(x)^2 in the denominator, so the reciprocal is squared. At x = pi/4 the wrong answer √2 is about 1.41, and the true slope is 2.

Using the derivative of arctan(x)

The derivative of arctan(x) is 1/(1 + x^2), a function that never exceeds 1 and is defined for every real x. sec(x)^2 is unbounded and has gaps at the asymptotes of tangent. Inverting a function does not invert its derivative.

Evaluating at x = pi/2

tan(pi/2) is undefined, so it has no slope there. sec(pi/2)^2 is undefined for the same reason: cosine is zero. The formula is only a derivative on the open intervals between consecutive asymptotes.

Derivative of arctan(x) — 1/(1 + x^2). The inverse tangent, not this derivative read backwards.
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 sec(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.