Step 1: Identify the function f(x) = x^3 - 3x^2 + 4.
Step 2: Find the derivative of the function, which is f'(x).
Step 3: Use the power rule to differentiate each term: The derivative of x^3 is 3x^2, the derivative of -3x^2 is -6x, and the derivative of the constant 4 is 0.
Step 4: Combine the derivatives to get f'(x) = 3x^2 - 6x.
Step 5: Substitute x = 2 into the derivative: f'(2) = 3(2^2) - 6(2).
Step 6: Calculate 2^2, which is 4, then multiply by 3 to get 12.
Step 7: Calculate 6(2), which is 12.
Step 8: Subtract the two results: 12 - 12 = 0.
Step 9: Conclude that f'(2) is equal to 0.
Differentiation – The process of finding the derivative of a function, which represents the rate of change of the function with respect to its variable.
Evaluation of Derivatives – Substituting a specific value into the derivative function to find the slope of the tangent line at that point.
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