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

Options

A. ` (1)/( xlog ( logx) ) `
B. ` (logx )/( xlog ( logx) ) `
C. ` (1 )/( x(logx) log ( logx) ) `
D. ` (1 )/( (logx) log ( logx) ) `

Answer

To find \(\frac{dy}{dx}\) for the function \(y = \log(\log(\log x))\), we will use the chain rule of differentiation. Here’s the step-by-step solution:

### Step 1: Identify the outer and inner functions
We can express \(y\) as a composition of functions:
- Let \(u = \log x\)
- Let \(v = \log u = \log(\log x)\)
- Then, \(y = \log v = \log(\log(\log x))\)

### Step 2: Differentiate using the chain rule
Using the chain rule, we have:
\[ \frac{dy}{dx} = \frac{dy}{dv} \cdot \frac{dv}{du} \cdot \frac{du}{dx} \]

### Step 3: Differentiate \(y\) with respect to \(v\)
The derivative of \(y\) with respect to \(v\) is:
\[ \frac{dy}{dv} = \frac{1}{v} = \frac{1}{\log(\log x)} \]

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

### Step 5: Differentiate \(u\) with respect to \(x\)
The derivative of \(u\) with respect to \(x\) is:
\[ \frac{du}{dx} = \frac{1}{x} \]

### Step 6: Combine the derivatives
Now, substituting back into the chain rule:
\[ \frac{dy}{dx} = \frac{1}{\log(\log x)} \cdot \frac{1}{\log x} \cdot \frac{1}{x} \]

### Step 7: Simplify the expression
Thus, we have:
\[ \frac{dy}{dx} = \frac{1}{x \cdot \log x \cdot \log(\log x)} \]

### Final Answer
\[ \frac{dy}{dx} = \frac{1}{x \log x \log(\log x)} \]

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