Given that `y=x^(5)+x^(4)+7`. Find `(dy)/(dx)`.

Options

A. `5x^(4)+4x^(3)`.
B. `5x^(3)+4x^(3)`.
C. `2x^(4)+4x^(3)`.
D. `5x^(4)+4x^(4)`.

Explanation

To find the derivative of the function \( y = x^5 + x^4 + 7 \) with respect to \( x \), we will differentiate each term individually. Here are the steps:

### Step 1: Identify the function
The function given is:
\[ y = x^5 + x^4 + 7 \]

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

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

2. **Differentiate \( x^4 \)**:
Again using the power rule:
\[ \frac{d}{dx}(x^4) = 4x^{4-1} = 4x^3 \]

3. **Differentiate the constant \( 7 \)**:
The derivative of a constant is zero:
\[ \frac{d}{dx}(7) = 0 \]

### Step 3: Combine the derivatives
Now, we combine the results from the differentiation of each term:
\[ \frac{dy}{dx} = 5x^4 + 4x^3 + 0 \]

### Step 4: Simplify the expression
Since adding zero does not change the value, we can simplify the expression:
\[ \frac{dy}{dx} = 5x^4 + 4x^3 \]

### Final Answer
Thus, the derivative of the function \( y \) with respect to \( x \) is:
\[ \frac{dy}{dx} = 5x^4 + 4x^3 \]

---