Question

If \( y = \tan x \), then at \( x = \frac{\pi}{4} \), \[ \frac{dy}{dx} = \quad ......... \]

Hint:

The derivative of \( \tan x \) is \( \sec^2 x \), and at \( x = \frac{\pi}{4} \), \( \sec^2 \frac{\pi}{4} = 2 \).

Answer 1

Step 1: Differentiating the function.
The derivative of \( y = \tan x \) is: \[ \frac{dy}{dx} = \sec^2 x \] Step 2: Substituting the value of \( x = \frac{\pi}{4} \).
At \( x = \frac{\pi}{4} \), we know that: \[ \sec \frac{\pi}{4} = \sqrt{2} \] So, \[ \frac{dy}{dx} = \sec^2 \frac{\pi}{4} = 2 \] Step 3: Conclusion.
Thus, the value of \( \frac{dy}{dx} \) at \( x = \frac{\pi}{4} \) is \( 2 \).