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.

Learn how to differentiate the function f(x) = x^7 + x^6 + x using the power rule. This calculus problem provides a step-by-step breakdown of applying the power rule to each term. Ideal for students in Grades 10-12 studying basic differentiation concepts.

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