To calculate the derivative of the function $f(x) = 7x + x^2$, we'll apply the basic rules of differentiation.

Step 1: Differentiate $7x$

The derivative of $7x$ is simply 7 because the derivative of a constant multiplied by $x$ is just the constant.

Step 2: Differentiate $x^2$

The derivative of $x^2$ is $2x$, according to the power rule: $\frac{d}{dx}(x^n) = n x^{n-1}$ where $n = 2$ in this case.

Final Result:

The derivative of $f(x) = 7x + x^2$ is: $f'(x) = 7 + 2x$

Would you like a more detailed explanation of the steps or have any further questions?

Here are some related questions for you:

1. How do you apply the power rule to differentiate $x^n$?
2. What happens if the function includes constants multiplied by $x^n$?
3. How do you differentiate more complex functions, such as trigonometric or exponential functions?
4. What are higher-order derivatives, and how do you compute them?
5. How can you apply the chain rule for composite functions?

Tip: Remember to apply the power rule carefully, especially with negative exponents or fractional powers.