To find the derivative of the function $f(x) = x^7 + x^6 + x$, we'll apply the power rule to each term. The power rule states that if $f(x) = x^n$, then $f'(x) = n \cdot x^{n-1}$.

Step-by-step differentiation:

1. For $x^7$, the derivative is $7x^6$.
2. For $x^6$, the derivative is $6x^5$.
3. For $x$, the derivative is $1$ (since the derivative of $x^1$ is just 1).

Now, combining all the results, we get:

$f'(x) = 7x^6 + 6x^5 + 1$

Would you like any additional details or have further questions?

Here are 5 related questions to deepen your understanding:

1. What is the derivative of $f(x) = x^8 + x^7 + x^2$?
2. How would you find the second derivative of $f(x) = x^7 + x^6 + x$?
3. Can you apply the product rule to a function like $f(x) = (x^7)(x^6)$?
4. How does the power rule differ from the chain rule in differentiation?
5. What would the integral of $f(x) = x^7 + x^6 + x$ be?

Tip: When differentiating, always look for whether the power rule or another rule (product, quotient, or chain rule) is required to simplify your work.