The differential coefficient of `f(x)=log(logx)` with respect to x is

Options

A. `(x)/(logx)`
B. `(logx)/(x)`
C. `(xlogx)^(-1)`
D. `xlogx`

Answer

To find the differential coefficient of the function \( f(x) = \log(\log x) \) with respect to \( x \), we will follow these steps:

### Step 1: Identify the function to differentiate
We have:
\[ f(x) = \log(\log x) \]

### Step 2: Apply the chain rule
To differentiate \( f(x) \), we will use the chain rule. The chain rule states that if you have a composite function \( g(h(x)) \), then the derivative is given by:
\[ \frac{dg}{dx} = \frac{dg}{dh} \cdot \frac{dh}{dx} \]
In our case, let:
- \( g(u) = \log(u) \) where \( u = \log x \)

Thus, we need to differentiate \( g(u) \) with respect to \( u \) and \( u \) with respect to \( x \):
\[ \frac{df}{dx} = \frac{dg}{du} \cdot \frac{du}{dx} \]

### Step 3: Differentiate \( g(u) = \log(u) \)
The derivative of \( g(u) \) with respect to \( u \) is:
\[ \frac{dg}{du} = \frac{1}{u} \]

### Step 4: Differentiate \( u = \log x \)
The derivative of \( u \) with respect to \( x \) is:
\[ \frac{du}{dx} = \frac{1}{x} \]

### Step 5: Combine the derivatives using the chain rule
Now, substituting back into our chain rule expression:
\[ \frac{df}{dx} = \frac{1}{\log x} \cdot \frac{1}{x} \]
This simplifies to:
\[ \frac{df}{dx} = \frac{1}{x \log x} \]

### Step 6: Final expression
Thus, the differential coefficient of \( f(x) = \log(\log x) \) with respect to \( x \) is:
\[ \frac{df}{dx} = \frac{1}{x \log x} \]