Derivative of tan(x)^2

Answer

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

Derivative of tan(x)^2

Rule Used: Chain Rule
Steps: Identify a composite function in $\tan(x)^2$ : use the chain rule $\frac{d}{dx} f(g(x))$ = f'(g(x))·g'(x). The inner derivative is required because a stretch or shift of the input changes how fast the output moves.
Differentiate the outer function, then multiply by the inner derivative.
Final answer: $2 \cdot s$ ec(x)^ $2 \cdot t$ an(x)

d/dx tan^2(x) = 2*sec(x)^2*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 tan(x)^2

Derivative TrigonometricIntermediate

Share this result

Share on X

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

Derivative

tan^2(x) → 2*sec(x)^2*tan(x)engine: 2*sec(x)^2*tan(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

tan^2(x) is an outer square whose inside derivative is sec^2(x), not 1. The chain rule multiplies 2 tan(x) by sec^2(x). Both factors are required: the 2 from the square, and sec^2 because tangent does not have derivative 1 except at isolated points. The result is undefined wherever cosine is zero, same as tangent itself.

Apply the chain rule to tan^2(x). The form to remember, already simplified, is 2*sec(x)^2*tan(x).

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

At x = 0.2, substitute into the simplified form 2*sec(x)^2*tan(x). That number is the slope of tan^2(x) there; a secant over a small step in x lands next to it.

Worked derivative

Write f(x) = tan^2(x).
Apply the chain rule to tan^2(x). The form to remember, already simplified, is 2*sec(x)^2*tan(x).
At x = 0.2, substitute into the simplified form 2*sec(x)^2*tan(x). That number is the slope of tan^2(x) there; a secant over a small step in x lands next to it.

Common mistakes

Answering 2*tan(x)

2 tan(x) would be correct if the inside were x. Tangent's derivative is sec^2, and at x = 0.2 that factor is already bigger than 1.

Dropping the inside derivative of tan^2(x)

If the inside is not exactly x, its derivative multiplies the outside derivative. Copying a basic trig formula onto a composed inside misses that factor.

Chain rule — The rule page lists this example and the others in the same family.
Trigonometric derivatives — 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(2x) — Same family of differentiation, different function.
Derivative of cos(2x) — Same family of differentiation, different function.
Derivative of sin^2(x) — Same family of differentiation, different function.
Derivative of cos^2(x) — Same family of differentiation, different function.

Frequently asked questions

What is the derivative of tan^2(x)?

2*sec(x)^2*tan(x). tan^2(x) is an outer square whose inside derivative is sec^2(x), not 1. The chain rule multiplies 2 tan(x) by sec^2(x). Both factors are required: the 2 from the square, and sec^2 because tangent does not have derivative 1 except at isolated points. The result is undefined wherever cosine is zero, same as tangent itself.