Differentiate sec x

June 3, 2026Mathematics

ScanSolve AI Answer

June 3, 2026

Answer

\sec x \tan x

Let's differentiate $\sec x$ .

Step 1: Rewrite $\sec x$ in terms of $\cos x$ . $\frac{d}{dx} (\sec x) = \frac{d}{dx} \left( \frac{1}{\cos x} \right) = \frac{d}{dx} ((\cos x)^{-1})$

Step 2: Apply the chain rule. Let $u = \cos x$ . Then $\frac{du}{dx} = -\sin x$ . The derivative of $u^{-1}$ with respect to $u$ is $-1 \cdot u^{-2}$ . So, using the chain rule $\frac{d}{dx} (u^{-1}) = \frac{d}{du} (u^{-1}) \cdot \frac{du}{dx}$ : $\frac{d}{dx} ((\cos x)^{-1}) = -1 \cdot (\cos x)^{-2} \cdot (-\sin x)$

Step 3: Simplify the expression. $-1 \cdot (\cos x)^{-2} \cdot (-\sin x) = \frac{\sin x}{(\cos x)^2} = \frac{\sin x}{\cos x} \cdot \frac{1}{\cos x}$ $= \tan x \cdot \sec x$

The derivative of $\sec x$ is $\boxed{\sec x \tan x}$ .

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Quick Answer

Rewrite x in terms of x. (d)/(dx) ( x) = (d)/(dx) ( (1)/( x) ) = (d)/(dx) (( x)^-1) Step 2: Apply the chain rule.

Asked on June 3, 2026Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

ScanSolve AI AnswerAnswered on June 3, 2026

Let's differentiate x. Step 1: Rewrite x in terms of x. (d)/(dx) ( x) = (d)/(dx) ( (1)/( x) ) = (d)/(dx) (( x)^-1) Step 2: Apply the chain rule. Let u = x. Then (du)/(dx) = - x. The derivative of u^-1 with respect to u is -1 · u^-2. So, using the chain rule (d)/(dx) (u^-1) = (d)/(du) (u^-1) · (du)/(dx): (d)/(dx) (( x)^-1) = -1 · ( x)^-2 · (- x) Step 3: Simplify the expression. -1 · ( x)^-2 · (- x) = ( x)/(( x)^2) = ( x)/( x) · (1)/( x) = x · x The derivative of x is x x. 3 done, 2 left today. You're making progress.