Question

\(\frac{d}{dx} \left(\sec^2 x - \tan^2 x\right)\)

• A. \( 2\sec^2 x - 2\tan x \)
• B. \( 2 \sec x - 2 \tan x \)
• C. 1
• D. 0

Hint:

If two terms are identical but have opposite signs in differentiation, they cancel each other out.

Answer 1

The Correct Option is D

We are tasked with finding the derivative of the function: \[ f(x) = \sec^2 x - \tan^2 x \] We will use the standard derivative formulas for \( \sec^2 x \) and \( \tan^2 x \) and apply differentiation rules. Step 1: Differentiate \( \sec^2 x \)
The derivative of \( \sec^2 x \) is well known: \[ \frac{d}{dx} \sec^2 x = 2 \sec^2 x \cdot \tan x \]
Step 2: Differentiate \( \tan^2 x \)
Similarly, the derivative of \( \tan^2 x \) is: \[ \frac{d}{dx} \tan^2 x = 2 \tan x \cdot \sec^2 x \]
Step 3: Combine the results
Now we subtract the two derivatives: \[ \frac{d}{dx} \left(\sec^2 x - \tan^2 x \right) = 2 \sec^2 x \cdot \tan x - 2 \tan x \cdot \sec^2 x \] Notice that both terms are identical but with opposite signs, so the expression simplifies to: \[ = 0 \] Thus, the derivative of \( \sec^2 x - \tan^2 x \) is 0.