Derivative of ` log[log(log x^(5))]` with respect to x is

Options

A. `(1)/(x log x log (log x ^(5)))`
B. `(1)/(x log (log x^(5)))`
C. `(5)/(x log (log x^(5)))`
D. None of these

Explanation

To find the derivative of the function \( y = \log(\log(\log(x^5))) \) with respect to \( x \), we will use the chain rule of differentiation. Here is the step-by-step solution:

### Step 1: Rewrite the function
We start with the function:
\[ y = \log(\log(\log(x^5))) \]

### Step 2: Differentiate using the chain rule
Using the chain rule, we differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = \frac{1}{\log(\log(x^5))} \cdot \frac{d}{dx}[\log(\log(x^5))] \]

### Step 3: Differentiate the inner function
Next, we need to differentiate \( \log(\log(x^5)) \):
\[ \frac{d}{dx}[\log(\log(x^5))] = \frac{1}{\log(x^5)} \cdot \frac{d}{dx}[\log(x^5)] \]

### Step 4: Differentiate \( \log(x^5) \)
Now we differentiate \( \log(x^5) \):
\[ \frac{d}{dx}[\log(x^5)] = \frac{5}{x} \]
Thus, we have:
\[ \frac{d}{dx}[\log(\log(x^5))] = \frac{1}{\log(x^5)} \cdot \frac{5}{x} \]

### Step 5: Substitute back into the derivative
Substituting this back into our expression for \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{1}{\log(\log(x^5))} \cdot \left(\frac{5}{x \log(x^5)}\right) \]

### Step 6: Simplify the expression
We can simplify this further:
\[ \frac{dy}{dx} = \frac{5}{x \log(x^5) \log(\log(x^5))} \]

### Step 7: Simplify \( \log(x^5) \)
Recall that \( \log(x^5) = 5 \log(x) \), so we substitute this into our expression:
\[ \frac{dy}{dx} = \frac{5}{x \cdot 5 \log(x) \log(\log(x^5))} \]
This simplifies to:
\[ \frac{dy}{dx} = \frac{1}{x \log(x) \log(\log(x^5))} \]

### Final Result
Thus, the derivative of \( y = \log(\log(\log(x^5))) \) with respect to \( x \) is:
\[ \frac{dy}{dx} = \frac{1}{x \log(x) \log(\log(x^5))} \]

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