Question

If $y = \log \sqrt{\tan x}$, then the value of $\frac{dy}{dx}$ at $x = \frac{\pi}{4}$ is

• A. 1
• B. $-1$
• C. $\frac{1}{2}$
• D. 0

Hint:

To make this derivative even simpler, convert the expression using basic trigonometric definitions: $\frac{\sec^2 x}{2\tan x} = \frac{1/\cos^2 x}{2(\sin x/\cos x)} = \frac{1}{2\sin x\cos x} = \frac{1}{\sin 2x} = \csc 2x$. Evaluating $\csc(2 \cdot \frac{\pi}{4}) = \csc(\frac{\pi}{2})$ instantly yields $1$.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The problem asks for the numerical value of the first derivative $\frac{dy}{dx}$ of a logarithmic composite function evaluated at the specific angle $x = \frac{\pi}{4}$.

Step 2: Key Formula or Approach:
1. Simplify the logarithmic function using the power rule: $\log(u^{1/2}) = \frac{1}{2}\log u$.
2. Differentiate the function using the chain rule: $$\frac{d}{dx}(\log(\tan x)) = \frac{1}{\tan x} \cdot \frac{d}{dx}(\tan x) = \frac{\sec^2 x}{\tan x}$$ 3. Substitute $x = \frac{\pi}{4}$ into the simplified derivative expression.

Step 3: Detailed Explanation:
Given function: $$y = \log \sqrt{\tan x} = \log(\tan x)^{1/2}$$ Using logarithmic properties, pull the power out to the front: $$y = \frac{1}{2} \log(\tan x)$$ Now, differentiate both sides with respect to $x$ using the chain rule: $$\frac{dy}{dx} = \frac{1}{2} \cdot \frac{1}{\tan x} \cdot \sec^2 x = \frac{\sec^2 x}{2\tan x}$$ We need to evaluate this derivative at $x = \frac{\pi}{4}$. We know that: $$\tan\left(\frac{\pi}{4}\right) = 1 \quad \text{and} \quad \sec\left(\frac{\pi}{4}\right) = \sqrt{2} \implies \sec^2\left(\frac{\pi}{4}\right) = 2$$ Substitute these trigonometric values into the derivative expression: $$\left. \frac{dy}{dx} \right|_{x=\frac{\pi}{4}} = \frac{2}{2(1)} = 1$$

Step 4: Final Answer:
The value of the derivative is 1, which corresponds to option (A).