Find `dy/dx if y= e^(5x)`

Answer

To find \( \frac{dy}{dx} \) when \( y = e^{5x} \), we will differentiate the function step by step.

### Step-by-Step Solution:

1. **Identify the function**:
We have \( y = e^{5x} \).

2. **Recall the differentiation rule for exponential functions**:
The derivative of \( e^{u} \) with respect to \( x \) is given by \( e^{u} \cdot \frac{du}{dx} \), where \( u \) is a function of \( x \).

3. **Set \( u = 5x \)**:
Here, \( u = 5x \).

4. **Differentiate \( u \)**:
We need to find \( \frac{du}{dx} \):
\[ \frac{du}{dx} = 5 \]

5. **Apply the differentiation rule**:
Now, we can differentiate \( y \):
\[ \frac{dy}{dx} = e^{u} \cdot \frac{du}{dx} = e^{5x} \cdot 5 \]

6. **Simplify the expression**:
Therefore, we have:
\[ \frac{dy}{dx} = 5e^{5x} \]

### Final Answer:
\[ \frac{dy}{dx} = 5e^{5x} \]