Find `dy/dx if y= 5x+x^5`

Answer

To find \( \frac{dy}{dx} \) for the function \( y = 5x + x^5 \), we will differentiate the function with respect to \( x \). Here are the steps:

### Step 1: Write down the function
We start with the given function:
\[ y = 5x + x^5 \]

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

1. **Differentiate \( 5x \)**:
- The derivative of \( 5x \) is \( 5 \) because the derivative of \( x \) with respect to \( x \) is \( 1 \) and \( 5 \) is a constant.

2. **Differentiate \( x^5 \)**:
- Using the power rule, the derivative of \( x^n \) is \( nx^{n-1} \). Here, \( n = 5 \), so:
\[ \frac{d}{dx}(x^5) = 5x^{5-1} = 5x^4 \]

### Step 3: Combine the derivatives
Now we can combine the derivatives from both terms:
\[ \frac{dy}{dx} = 5 + 5x^4 \]

### Final Answer
Thus, the derivative \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = 5 + 5x^4 \]