If f(x) = x^4 - 4x^3, find f'(2). (2023)

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Question: If f(x) = x^4 - 4x^3, find f\'(2). (2023)

Options:

Correct Answer: 16

Exam Year: 2023

Solution:

f\'(x) = 4x^3 - 12x^2; thus, f\'(2) = 4(2^3) - 12(2^2) = 32 - 48 = -16.

If f(x) = x^4 - 4x^3, find f'(2). (2023)

If f(x) = x^4 - 4x^3, find f'(2). (2023)

Step 1: Identify the function f(x) = x^4 - 4x^3.
Step 2: Find the derivative f'(x) using the power rule. The power rule states that if f(x) = x^n, then f'(x) = n*x^(n-1).
Step 3: Apply the power rule to each term in f(x):
- For x^4, the derivative is 4*x^(4-1) = 4x^3.
- For -4x^3, the derivative is -4*3*x^(3-1) = -12x^2.
Step 4: Combine the derivatives to get f'(x) = 4x^3 - 12x^2.
Step 5: Now, substitute x = 2 into the derivative f'(x).
Step 6: Calculate f'(2) = 4(2^3) - 12(2^2).
Step 7: Calculate 2^3 = 8, so 4(2^3) = 4*8 = 32.
Step 8: Calculate 2^2 = 4, so 12(2^2) = 12*4 = 48.
Step 9: Now, subtract: 32 - 48 = -16.
Step 10: Therefore, f'(2) = -16.

Differentiation – The process of finding the derivative of a function, which represents the rate of change of the function with respect to its variable.
Polynomial Functions – Understanding the behavior and properties of polynomial functions, including their derivatives.
Evaluation of Derivatives – Calculating the value of the derivative at a specific point.