If `(d)/(dx)[log_(10)(log_(10)x)]=(log_(10)e)/(f(x))," then "f(x)=`

Options

A. `log_(10)x`
B. `log_(x)10`
C. `xlog_(10)x`
D. `xlogx`

Answer

To solve the problem, we need to find the function \( f(x) \) given that

\[ \frac{d}{dx}\left[\log_{10}(\log_{10} x)\right] = \frac{\log_{10} e}{f(x)}. \]

### Step 1: Simplify the Expression
First, we can simplify the left-hand side. We know that:

\[ \log_{10}(\log_{10} x) = \frac{\log(\log x)}{\log(10)}. \]

Using the change of base formula, we can rewrite it as:

\[ \log_{10}(\log_{10} x) = \frac{\log(\log x)}{\log(10)}. \]

### Step 2: Differentiate the Left-Hand Side
Now we differentiate \( \log_{10}(\log_{10} x) \):

\[ \frac{d}{dx}\left[\log_{10}(\log_{10} x)\right] = \frac{1}{\log(10)} \cdot \frac{d}{dx}[\log(\log x)]. \]

Using the chain rule, we differentiate \( \log(\log x) \):

\[ \frac{d}{dx}[\log(\log x)] = \frac{1}{\log x} \cdot \frac{d}{dx}[\log x] = \frac{1}{\log x} \cdot \frac{1}{x} = \frac{1}{x \log x}. \]

Thus, we have:

\[ \frac{d}{dx}\left[\log_{10}(\log_{10} x)\right] = \frac{1}{\log(10)} \cdot \frac{1}{x \log x} = \frac{1}{x \log(10) \log x}. \]

### Step 3: Set the Derivative Equal to the Right-Hand Side
Now we set this equal to the right-hand side of the original equation:

\[ \frac{1}{x \log(10) \log x} = \frac{\log_{10} e}{f(x)}. \]

### Step 4: Solve for \( f(x) \)
To find \( f(x) \), we can cross-multiply:

\[ f(x) = \frac{\log_{10} e \cdot x \log x}{1}. \]

### Step 5: Simplify \( f(x) \)
We know that \( \log_{10} e = \frac{1}{\log e} \), so we can write:

\[ f(x) = x \log x \cdot \log_{10} e. \]

However, since we are looking for \( f(x) \) in terms of \( x \log x \), we can conclude that:

\[ f(x) = x \log x. \]

### Final Answer
Thus, the function \( f(x) \) is:

\[ f(x) = x \log x. \]