Find `dy/dx if e^x=logy`

Answer

To find \( \frac{dy}{dx} \) given the equation \( e^x = \log y \), we will follow these steps:

### Step 1: Differentiate both sides with respect to \( x \)
We start with the equation:
\[ e^x = \log y \]
Now, we differentiate both sides with respect to \( x \).

### Step 2: Differentiate the left side
The left side is \( e^x \). The derivative of \( e^x \) with respect to \( x \) is:
\[ \frac{d}{dx}(e^x) = e^x \]

### Step 3: Differentiate the right side
The right side is \( \log y \). Using the chain rule, the derivative of \( \log y \) with respect to \( x \) is:
\[ \frac{d}{dx}(\log y) = \frac{1}{y} \cdot \frac{dy}{dx} \]

### Step 4: Set the derivatives equal
Now we set the derivatives from both sides equal to each other:
\[ e^x = \frac{1}{y} \cdot \frac{dy}{dx} \]

### Step 5: Solve for \( \frac{dy}{dx} \)
To isolate \( \frac{dy}{dx} \), we multiply both sides by \( y \):
\[ y \cdot e^x = \frac{dy}{dx} \]
Thus, we have:
\[ \frac{dy}{dx} = y \cdot e^x \]

### Final Result
The final expression for \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = y \cdot e^x \]