Question

If \( y = \tan x \), then at \( x = \frac{\pi}{4} \), \( \frac{dy}{dx} \) is equal to:

• A. 1
• B. \( \sqrt{2} \)
• C. 2
• D. 4

Hint:

Memorize the derivatives of all trigonometric functions. \( \frac{d}{dx}(\tan x) = \sec^2 x \) and \( \frac{d}{dx}(\cot x) = -\operatorname{cosec}^2 x \).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
We need to find the derivative of the tangent function and evaluate it at a specific point.
Step 2: Detailed Explanation:
Given: \( y = \tan x \).
Differentiating with respect to \( x \):
\[ \frac{dy}{dx} = \sec^2 x \]
We need the value at \( x = \frac{\pi}{4} \):
\[ \left[ \frac{dy}{dx} \right]_{x = \frac{\pi}{4}} = \sec^2\left(\frac{\pi}{4}\right) \] We know that \( \cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \), so \( \sec\left(\frac{\pi}{4}\right) = \sqrt{2} \).
Substituting this:
\[ = (\sqrt{2})^2 = 2 \] Step 3: Final Answer:
The value of the derivative at \( x = \frac{\pi}{4} \) is 2.