If `2^(x)=3^(y)`, then find `dy/dx`.

Answer

To find \( \frac{dy}{dx} \) given the equation \( 2^x = 3^y \), we can follow these steps:

### Step 1: Take the natural logarithm of both sides
We start by taking the natural logarithm of both sides of the equation:

\[ \ln(2^x) = \ln(3^y) \]

### Step 2: Use the logarithmic identity
Using the property of logarithms that states \( \ln(a^b) = b \ln(a) \), we can rewrite the equation:

\[ x \ln(2) = y \ln(3) \]

### Step 3: Differentiate both sides with respect to \( x \)
Now we differentiate both sides of the equation with respect to \( x \):

\[ \frac{d}{dx}(x \ln(2)) = \frac{d}{dx}(y \ln(3)) \]

Using the product rule on the left side and the chain rule on the right side, we get:

\[ \ln(2) = \ln(3) \frac{dy}{dx} \]

### Step 4: Solve for \( \frac{dy}{dx} \)
Now, we can isolate \( \frac{dy}{dx} \):

\[ \frac{dy}{dx} = \frac{\ln(2)}{\ln(3)} \]

### Final Answer
Thus, the derivative \( \frac{dy}{dx} \) is:

\[ \frac{dy}{dx} = \frac{\ln(2)}{\ln(3)} \]

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