Differentiate `x^5 + e^(x)` w.r.t `x`

Answer

To differentiate the function \( y = x^5 + e^x \) with respect to \( x \), we will follow these steps:

### Step 1: Identify the function
We start with the function:
\[ y = x^5 + e^x \]

### Step 2: Differentiate each term
We will differentiate the function term by term.

1. **Differentiate \( x^5 \)**:
Using the power rule of differentiation, which states that if \( y = x^n \), then \( \frac{dy}{dx} = n \cdot x^{n-1} \):
\[ \frac{d}{dx}(x^5) = 5 \cdot x^{5-1} = 5x^4 \]

2. **Differentiate \( e^x \)**:
The derivative of \( e^x \) is simply \( e^x \):
\[ \frac{d}{dx}(e^x) = e^x \]

### Step 3: Combine the derivatives
Now, we combine the results from the differentiation of each term:
\[ \frac{dy}{dx} = \frac{d}{dx}(x^5) + \frac{d}{dx}(e^x) = 5x^4 + e^x \]

### Final Answer
Thus, the derivative of the function \( y = x^5 + e^x \) with respect to \( x \) is:
\[ \frac{dy}{dx} = 5x^4 + e^x \]

---