Differentiate the following function with respect to `x`
`e^((5x+2))`

Options

A. `5e^(x+2)`
B. `e^(5x+2)`
C. `5e^(5x+2)`
D. `5e^(5x)`

Explanation

To differentiate the function \( e^{(5x+2)} \) with respect to \( x \), we will use the chain rule. The chain rule states that if you have a composite function \( f(g(x)) \), then the derivative is given by \( f'(g(x)) \cdot g'(x) \).

### Step-by-step Solution:

1. **Identify the outer and inner functions**:
- The outer function is \( f(u) = e^u \) where \( u = 5x + 2 \).
- The inner function is \( g(x) = 5x + 2 \).

2. **Differentiate the outer function**:
- The derivative of \( f(u) = e^u \) with respect to \( u \) is \( f'(u) = e^u \).

3. **Differentiate the inner function**:
- The derivative of \( g(x) = 5x + 2 \) with respect to \( x \) is \( g'(x) = 5 \).

4. **Apply the chain rule**:
- According to the chain rule, the derivative of \( e^{(5x+2)} \) is:
\[ \frac{d}{dx} e^{(5x+2)} = f'(g(x)) \cdot g'(x) = e^{(5x+2)} \cdot 5 \]

5. **Write the final answer**:
- Thus, the derivative of \( e^{(5x+2)} \) with respect to \( x \) is:
\[ 5e^{(5x+2)} \]

### Final Answer:
\[ \frac{d}{dx} e^{(5x+2)} = 5e^{(5x+2)} \]

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