To find the derivative of the function \( y = 4 + 5x + 7x^3 \) with respect to \( x \), we will differentiate each term of the function separately.
### Step-by-Step Solution:
1. **Identify the function**:
We have \( y = 4 + 5x + 7x^3 \).
2. **Differentiate each term**:
We will apply the differentiation rules to each term in the function.
- The derivative of a constant (4) is 0.
- The derivative of \( 5x \) is \( 5 \) (since the derivative of \( x \) is 1).
- The derivative of \( 7x^3 \) can be found using the power rule:
\[
\frac{d}{dx}(x^n) = n \cdot x^{n-1}
\]
Here, \( n = 3 \), so:
\[
\frac{d}{dx}(7x^3) = 7 \cdot 3x^{3-1} = 21x^2
\]
3. **Combine the derivatives**:
Now, we can combine the results from the differentiation of each term:
\[
\frac{dy}{dx} = 0 + 5 + 21x^2
\]
4. **Simplify the expression**:
Thus, the final result for the derivative is:
\[
\frac{dy}{dx} = 5 + 21x^2
\]
### Final Answer:
\[
\frac{dy}{dx} = 5 + 21x^2
\]
To find the derivative of the function \( y = 4 + 5x + 7x^3 \) with respect to \( x \), we will differentiate each term of the function separately.
### Step-by-Step Solution:
1. **Identify the function**:
We have \( y = 4 + 5x + 7x^3 \).
2. **Differentiate each term**:
...
