If `y=e^([log (x+1)-log x])." Find" (dy)/(dx)`

Answer

To solve the problem, we need to find the derivative of the function \( y = e^{\log(x+1) - \log x} \).

### Step-by-Step Solution:

1. **Simplify the expression**:
We can use the properties of logarithms to simplify the exponent:
\[ \log(x+1) - \log x = \log\left(\frac{x+1}{x}\right) \]
Therefore, we can rewrite \( y \) as:
\[ y = e^{\log\left(\frac{x+1}{x}\right)} \]

2. **Use the property of exponentials and logarithms**:
Since \( e^{\log(a)} = a \), we can simplify \( y \) further:
\[ y = \frac{x+1}{x} \]

3. **Differentiate \( y \)**:
Now, we differentiate \( y \) with respect to \( x \):
\[ y = \frac{x+1}{x} = 1 + \frac{1}{x} \]
The derivative of \( y \) is:
\[ \frac{dy}{dx} = 0 - \frac{1}{x^2} = -\frac{1}{x^2} \]

4. **Final result**:
Thus, the derivative \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = -\frac{1}{x^2} \]