Math Problem Statement

Solution

To solve the problem of finding the derivative f(x)f'(x) for the given function f(x)=7x88x+2f(x) = 7x^8 - 8x + 2:

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

  • The first term is 7x87x^8. The derivative is found using the power rule ddx[xn]=nxn1\frac{d}{dx} [x^n] = n \cdot x^{n-1}. ddx[7x8]=78x81=56x7\frac{d}{dx} [7x^8] = 7 \cdot 8x^{8-1} = 56x^7

  • The second term is 8x-8x. The derivative of xx is 1. ddx[8x]=81=8\frac{d}{dx} [-8x] = -8 \cdot 1 = -8

  • The third term is a constant 22. The derivative of a constant is 0. ddx[2]=0\frac{d}{dx} [2] = 0

Step 2: Combine the derivatives

f(x)=56x78f'(x) = 56x^7 - 8

Step 3: Compare with the given options

The correct answer is C. f(x)=56x78f'(x) = 56x^7 - 8.


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 axnax^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.

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Math Problem Analysis

Mathematical Concepts

Differentiation
Power Rule
Derivative of Constants
Polynomial Functions

Formulas

Power rule for differentiation: \( \frac{d}{dx}[x^n] = n \cdot x^{n-1} \)

Theorems

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Suitable Grade Level

High School