If `y=log_(sinx)(tanx),` then `(dy)/ (dx)` at `x=(1)/(4)` is equal to

Options

A. `(4)/(log2)`
B. `-4log2`
C. `(-4)/(log2)`
D. None of these

Answer

To find the derivative \(\frac{dy}{dx}\) for the function \(y = \log_{\sin x}(\tan x)\) at \(x = \frac{\pi}{4}\), we can follow these steps:

### Step 1: Rewrite the logarithm
Using the change of base formula for logarithms, we can rewrite the function:
\[ y = \frac{\log(\tan x)}{\log(\sin x)} \]

### Step 2: Differentiate using the quotient rule
To differentiate \(y\) with respect to \(x\), we apply the quotient rule:
\[ \frac{dy}{dx} = \frac{g(x)f'(x) - f(x)g'(x)}{(g(x))^2} \]
where \(f(x) = \log(\tan x)\) and \(g(x) = \log(\sin x)\).

### Step 3: Find \(f'(x)\) and \(g'(x)\)
We need to find the derivatives \(f'(x)\) and \(g'(x)\):
- For \(f(x) = \log(\tan x)\):
\[ f'(x) = \frac{1}{\tan x} \cdot \sec^2 x = \frac{\sec^2 x}{\tan x} \]
- For \(g(x) = \log(\sin x)\):
\[ g'(x) = \frac{1}{\sin x} \cdot \cos x = \frac{\cos x}{\sin x} = \cot x \]

### Step 4: Substitute \(f(x)\), \(g(x)\), \(f'(x)\), and \(g'(x)\) into the quotient rule
Now substituting into the quotient rule:
\[ \frac{dy}{dx} = \frac{\log(\sin x) \cdot \frac{\sec^2 x}{\tan x} - \log(\tan x) \cdot \cot x}{(\log(\sin x))^2} \]

### Step 5: Evaluate at \(x = \frac{\pi}{4}\)
Now we substitute \(x = \frac{\pi}{4}\):
- \(\tan\left(\frac{\pi}{4}\right) = 1\)
- \(\sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}\)
- \(\sec^2\left(\frac{\pi}{4}\right) = 2\)
- \(\cot\left(\frac{\pi}{4}\right) = 1\)

Substituting these values:
\[ \frac{dy}{dx} = \frac{\log\left(\frac{1}{\sqrt{2}}\right) \cdot \frac{2}{1} - \log(1) \cdot 1}{\left(\log\left(\frac{1}{\sqrt{2}}\right)\right)^2} \]
Since \(\log(1) = 0\), the equation simplifies to:
\[ \frac{dy}{dx} = \frac{2 \log\left(\frac{1}{\sqrt{2}}\right)}{\left(\log\left(\frac{1}{\sqrt{2}}\right)\right)^2} \]

### Step 6: Simplify the expression
We know that \(\log\left(\frac{1}{\sqrt{2}}\right) = -\frac{1}{2} \log(2)\):
\[ \frac{dy}{dx} = \frac{2 \left(-\frac{1}{2} \log(2)\right)}{\left(-\frac{1}{2} \log(2)\right)^2} \]
This simplifies to:
\[ \frac{dy}{dx} = \frac{-\log(2)}{\frac{1}{4} (\log(2))^2} = -4 \frac{\log(2)}{(\log(2))^2} = -\frac{4}{\log(2)} \]

### Final Answer
Thus, the value of \(\frac{dy}{dx}\) at \(x = \frac{\pi}{4}\) is:
\[ \frac{dy}{dx} = -\frac{4}{\log(2)} \]