To solve the problem of finding the derivative $f'(x)$ for the given function $f(x) = 7x^8 - 8x + 2$:

Step 1: Differentiate each term of $f(x)$

• The first term is $7x^8$. The derivative is found using the power rule $\frac{d}{dx} [x^n] = n \cdot x^{n-1}$. $\frac{d}{dx} [7x^8] = 7 \cdot 8x^{8-1} = 56x^7$

• The second term is $-8x$. The derivative of $x$ is 1. $\frac{d}{dx} [-8x] = -8 \cdot 1 = -8$

• The third term is a constant $2$. The derivative of a constant is 0. $\frac{d}{dx} [2] = 0$

Step 2: Combine the derivatives

$f'(x) = 56x^7 - 8$

Step 3: Compare with the given options

The correct answer is C. $f'(x) = 56x^7 - 8$.

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Would you like more details on any step or have any questions? Here are some related questions you might be interested in:

1. What is the power rule for differentiation?
2. How do you differentiate a constant?
3. How do you apply the derivative rules to functions with multiple terms?
4. What is the general rule for differentiating $ax^n$?
5. How do you handle negative exponents in differentiation?
6. How do you differentiate polynomial functions?
7. What is the importance of simplifying after differentiation?
8. How does the chain rule differ from the power rule in differentiation?

Tip: When differentiating polynomials, always apply the power rule term by term and combine the results.