What is the derivative of f(x) = x^4 + 2x^3 - x + 1? (2017)

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Question: What is the derivative of f(x) = x^4 + 2x^3 - x + 1? (2017)

Options:

Correct Answer: 4x^3 + 6x^2 - 1

Exam Year: 2017

Solution:

The derivative f\'(x) = d/dx(x^4 + 2x^3 - x + 1) = 4x^3 + 6x^2 - 1.

What is the derivative of f(x) = x^4 + 2x^3 - x + 1? (2017)

What is the derivative of f(x) = x^4 + 2x^3 - x + 1? (2017)

Step 1: Identify the function you want to differentiate, which is f(x) = x^4 + 2x^3 - x + 1.
Step 2: Recall the power rule for differentiation: if f(x) = x^n, then f'(x) = n*x^(n-1).
Step 3: Differentiate each term in the function separately using the power rule.
Step 4: For the first term x^4, apply the power rule: the derivative is 4*x^(4-1) = 4x^3.
Step 5: For the second term 2x^3, apply the power rule: the derivative is 2*3*x^(3-1) = 6x^2.
Step 6: For the third term -x, remember that -x is the same as -1*x^1. The derivative is -1*1*x^(1-1) = -1.
Step 7: For the constant term +1, the derivative is 0 because the derivative of any constant is 0.
Step 8: Combine all the derivatives from the previous steps: 4x^3 + 6x^2 - 1 + 0.
Step 9: Write the final answer for the derivative: f'(x) = 4x^3 + 6x^2 - 1.

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