If f(x) = x^4 - 2x^3 + x, what is f'(1)? (2023)

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Question: If f(x) = x^4 - 2x^3 + x, what is f\'(1)? (2023)

Options:

Correct Answer: 2

Exam Year: 2023

Solution:

First, find f\'(x) = 4x^3 - 6x^2 + 1. Then, f\'(1) = 4(1)^3 - 6(1)^2 + 1 = 4 - 6 + 1 = -1.

If f(x) = x^4 - 2x^3 + x, what is f'(1)? (2023)

If f(x) = x^4 - 2x^3 + x, what is f'(1)? (2023)

Step 1: Identify the function f(x) = x^4 - 2x^3 + x.
Step 2: Find the derivative of the function, f'(x). To do this, apply the power rule to each term.
Step 3: The derivative of x^4 is 4x^3.
Step 4: The derivative of -2x^3 is -6x^2.
Step 5: The derivative of x is 1.
Step 6: Combine the derivatives to get f'(x) = 4x^3 - 6x^2 + 1.
Step 7: Now, we need to find f'(1). Substitute x = 1 into the derivative f'(x).
Step 8: Calculate f'(1) = 4(1)^3 - 6(1)^2 + 1.
Step 9: Simplify the expression: 4(1) - 6(1) + 1 = 4 - 6 + 1.
Step 10: Perform the arithmetic: 4 - 6 = -2, then -2 + 1 = -1.
Step 11: Therefore, f'(1) = -1.

Differentiation – The process of finding the derivative of a function.
Evaluation of Derivatives – Substituting a specific value into the derivative to find the slope at that point.