Differentiate log (7 log x) w.r.t.x

Answer

To differentiate the function \( y = \log(7 \log x) \) with respect to \( x \), we will use the chain rule and the properties of logarithms. Here’s a step-by-step solution:

### Step 1: Rewrite the function
We start with the function:
\[ y = \log(7 \log x) \]
Using the property of logarithms, we can rewrite this as:
\[ y = \log 7 + \log(\log x) \]

### Step 2: Differentiate the constant term
The derivative of a constant is zero, so:
\[ \frac{d}{dx}(\log 7) = 0 \]

### Step 3: Differentiate the logarithmic term
Now, we differentiate \( \log(\log x) \) using the chain rule. The derivative of \( \log u \) is \( \frac{1}{u} \cdot \frac{du}{dx} \). Here, \( u = \log x \):
\[ \frac{d}{dx}(\log(\log x)) = \frac{1}{\log x} \cdot \frac{d}{dx}(\log x) \]

### Step 4: Differentiate \( \log x \)
The derivative of \( \log x \) is:
\[ \frac{d}{dx}(\log x) = \frac{1}{x} \]

### Step 5: Combine the results
Substituting back into our derivative:
\[ \frac{d}{dx}(\log(\log x)) = \frac{1}{\log x} \cdot \frac{1}{x} \]
Thus, the total derivative of \( y \) is:
\[ \frac{dy}{dx} = 0 + \frac{1}{\log x} \cdot \frac{1}{x} = \frac{1}{x \log x} \]

### Final Answer
Therefore, the derivative of \( \log(7 \log x) \) with respect to \( x \) is:
\[ \frac{dy}{dx} = \frac{1}{x \log x} \]

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