If f(x) = x^4 - 4x^3 + 6x^2, find f'(2).

If f(x) = x^4 - 4x^3 + 6x^2, find f'(2).

Step 1: Write down the function f(x) = x^4 - 4x^3 + 6x^2.
Step 2: Find the derivative f'(x) by 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.
- For 6x^2, the derivative is 6*2*x^(2-1) = 12x.
Step 4: Combine the derivatives to get f'(x) = 4x^3 - 12x^2 + 12x.
Step 5: Now, substitute x = 2 into f'(x) to find f'(2).
Step 6: Calculate f'(2) = 4(2^3) - 12(2^2) + 12(2).
Step 7: Compute each term: 4(2^3) = 4*8 = 32, -12(2^2) = -12*4 = -48, and 12(2) = 24.
Step 8: Add the results together: 32 - 48 + 24 = 8.
Step 9: Therefore, f'(2) = 8.

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 properties and behavior of polynomial functions, including their derivatives.
Evaluation of Derivatives – Calculating the value of the derivative at a specific point.