If ` y = 2^("log x ") ` , then ` (dy)/(dt)` is

Options

A. `(2^(logx))/(log2)`
B. `2^(logx).log_(e)2`
C. `2^(logx)/(x) `
D. `(2^(logx).log_(e)2)/(x)`

Explanation

To solve the problem \( y = 2^{\log x} \) and find \( \frac{dy}{dt} \), we will follow these steps:

### Step 1: Rewrite the function
We start with the function:
\[ y = 2^{\log x} \]

### Step 2: Substitute \( t \) for \( \log x \)
Let:
\[ t = \log x \]
Then we can rewrite \( y \) as:
\[ y = 2^t \]

### Step 3: Differentiate \( y \) with respect to \( t \)
Using the differentiation rule for exponential functions, we have:
\[ \frac{dy}{dt} = 2^t \cdot \log 2 \]

### Step 4: Differentiate \( t \) with respect to \( x \)
Now we differentiate \( t \) with respect to \( x \):
\[ \frac{dt}{dx} = \frac{1}{x} \]

### Step 5: Use the chain rule to find \( \frac{dy}{dx} \)
Using the chain rule:
\[ \frac{dy}{dx} = \frac{dy}{dt} \cdot \frac{dt}{dx} \]
Substituting the values we found:
\[ \frac{dy}{dx} = (2^t \cdot \log 2) \cdot \frac{1}{x} \]
Since \( t = \log x \), we can substitute back:
\[ \frac{dy}{dx} = \frac{2^{\log x} \cdot \log 2}{x} \]

### Final Answer
Thus, the derivative \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = \frac{2^{\log x} \cdot \log 2}{x} \]

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