Find the value of the derivative of f(x) = x^4 - 4x^3 + 6x^2 at x = 1.

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Question: Find the value of the derivative of f(x) = x^4 - 4x^3 + 6x^2 at x = 1.

Options:

Correct Answer: 0

Solution:

f\'(x) = 4x^3 - 12x^2 + 12x. Evaluating at x = 1 gives f\'(1) = 4 - 12 + 12 = 4.

Find the value of the derivative of f(x) = x^4 - 4x^3 + 6x^2 at x = 1.

Find the value of the derivative of f(x) = x^4 - 4x^3 + 6x^2 at x = 1.

Correct Answer: 4

Step 1: Write down the function f(x) = x^4 - 4x^3 + 6x^2.
Step 2: Find the derivative of f(x). The derivative f'(x) is calculated using the power rule.
Step 3: Apply the power rule to each term: The derivative of x^4 is 4x^3, the derivative of -4x^3 is -12x^2, and the derivative of 6x^2 is 12x.
Step 4: Combine the derivatives to get f'(x) = 4x^3 - 12x^2 + 12x.
Step 5: Now, evaluate the derivative at x = 1. Substitute 1 into f'(x).
Step 6: Calculate f'(1) = 4(1)^3 - 12(1)^2 + 12(1).
Step 7: Simplify the expression: f'(1) = 4 - 12 + 12.
Step 8: Perform the arithmetic: 4 - 12 = -8, and -8 + 12 = 4.
Step 9: The value of the derivative at x = 1 is 4.

Differentiation – The process of finding the derivative of a function to determine its rate of change.
Evaluation of Derivatives – Substituting a specific value into the derivative to find the slope of the tangent line at that point.