The derivative of sec≤ft(tan√x) with respect to x is:

The correct answer is Option (1) → \(\displaystyle\frac{\sec\left(\tan\sqrt{x}\right)\tan\left(\tan\sqrt{x}\right)\sec^2\sqrt{x}}{2\sqrt{x}}\)

This is a function of a function of a function, so the chain rule is applied three times: sec, then tan, then the square root.

    \( \displaystyle \frac{d}{dx}\sec\left(\tan\sqrt{x}\right) = \sec\left(\tan\sqrt{x}\right)\tan\left(\tan\sqrt{x}\right) \cdot \frac{d}{dx}\tan\sqrt{x} \)

    \( \displaystyle \frac{d}{dx}\tan\sqrt{x} = \sec^2\sqrt{x} \cdot \frac{d}{dx}\sqrt{x} = \sec^2\sqrt{x} \cdot \frac{1}{2\sqrt{x}} \)

Putting these together:

    \( \displaystyle \frac{d}{dx}\sec\left(\tan\sqrt{x}\right) = \frac{\sec\left(\tan\sqrt{x}\right)\tan\left(\tan\sqrt{x}\right)\sec^2\sqrt{x}}{2\sqrt{x}} \)

Why the other options are incorrect

• Option (2) \(\sec^2\left(\tan\sqrt{x}\right)\) — Incorrect. The derivative of sec u is sec u tan u, not sec² u, and the inner factors are missing.

• Option (3) \(\displaystyle\frac{\sec\left(\tan\sqrt{x}\right)\tan\left(\tan\sqrt{x}\right)\sec^2\sqrt{x}}{x}\) — Incorrect. The derivative of \(\sqrt{x}\) is \(\frac{1}{2\sqrt{x}}\), not \(\frac{1}{x}\).

• Option (4) \(\sec^2\left(\tan x^{\frac{1}{3}}\right)\) — Incorrect. It has a cube root in place of the square root and none of the chain-rule factors.

NCERT (2026-27 edition):
Derivatives of Composite Functions