Question

If \[ \frac{d}{dx} \left( \frac{\sec x+\tan x} {\sec x-\tan x} \right) =k \] at \[ x=\frac{\pi}{4}, \] then \[ \frac{k}{2\sqrt2}-2\sqrt2= \ ?} \]

• A. \(5\sqrt2\)
• B. \(12\sqrt2\)
• C. \(3\)
• D. \(9\)

Hint:

Always look for trigonometric identities before differentiating complicated expressions.

Answer 1

The Correct Option is C

Concept: Use the identity \[ (\sec x+\tan x)(\sec x-\tan x)=1. \] This greatly simplifies the given function.

Step 1: Simplify the expression.
Since \[ \sec x-\tan x = \frac1{\sec x+\tan x}, \] we get \[ \frac{\sec x+\tan x} {\sec x-\tan x} = (\sec x+\tan x)^2. \] Let \[ y=(\sec x+\tan x)^2. \]

Step 2: Differentiate.
\[ \frac{dy}{dx} = 2(\sec x+\tan x) (\sec x\tan x+\sec^2x). \] At \[ x=\frac{\pi}{4}, \] \[ \sec\frac{\pi}{4}=\sqrt2, \qquad \tan\frac{\pi}{4}=1. \] Thus \[ k = 2(\sqrt2+1)(\sqrt2+2). \] \[ = 2(4+3\sqrt2). \] \[ = 8+6\sqrt2. \]

Step 3: Evaluate the required expression.
\[ \frac{k}{2\sqrt2} = \frac{8+6\sqrt2}{2\sqrt2}. \] \[ = 2\sqrt2+3. \] Therefore, \[ \frac{k}{2\sqrt2}-2\sqrt2 = 3. \] Hence \[ \boxed{3}. \]