If ` y= log _5 (log _7x),then (dy)/(dx) =`

Options

A. ` (1)/( xlog 5 *log _7x ) `
B. ` (1)/( xlog 5 *logx ) `
C. ` (1)/( xlog 7 *log_5x ) `
D. ` (1)/( xlog 7 *logx ) `

Answer

To find \(\frac{dy}{dx}\) for the function \(y = \log_5(\log_7 x)\), we will use the change of base formula for logarithms and then differentiate. Here’s the step-by-step solution:

### Step 1: Change of Base
We start with the function:
\[ y = \log_5(\log_7 x) \]
Using the change of base formula, we can express this in terms of natural logarithms:
\[ y = \frac{\log(\log_7 x)}{\log 5} \]

### Step 2: Change of Base for \(\log_7 x\)
Next, we apply the change of base formula again to \(\log_7 x\):
\[ \log_7 x = \frac{\log x}{\log 7} \]
Substituting this back into our expression for \(y\):
\[ y = \frac{\log\left(\frac{\log x}{\log 7}\right)}{\log 5} \]

### Step 3: Simplify the Logarithm
Using the property of logarithms \(\log(a/b) = \log a - \log b\):
\[ y = \frac{\log(\log x) - \log(\log 7)}{\log 5} \]

### Step 4: Differentiate \(y\)
Now we differentiate \(y\) with respect to \(x\):
\[ \frac{dy}{dx} = \frac{1}{\log 5} \cdot \frac{d}{dx}(\log(\log x) - \log(\log 7)) \]
Since \(\log(\log 7)\) is a constant, its derivative is 0:
\[ \frac{dy}{dx} = \frac{1}{\log 5} \cdot \frac{d}{dx}(\log(\log x)) \]

### Step 5: Differentiate \(\log(\log x)\)
Using the chain rule:
\[ \frac{d}{dx}(\log(\log x)) = \frac{1}{\log x} \cdot \frac{d}{dx}(\log x) = \frac{1}{\log x} \cdot \frac{1}{x} \]
Thus,
\[ \frac{dy}{dx} = \frac{1}{\log 5} \cdot \frac{1}{x \log x} \]

### Final Answer
Putting it all together, we have:
\[ \frac{dy}{dx} = \frac{1}{x \log 5 \log x} \]