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

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

Options:

Correct Answer: 4x^3 - 12x^2 + 12x

Solution:

Using the power rule, f\'(x) = 4x^3 - 12x^2 + 12x.

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

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

Step 1: Identify the function you want to differentiate, which is f(x) = x^4 - 4x^3 + 6x^2 - 2.
Step 2: Recall the power rule for differentiation. The power rule states that if you have x^n, the derivative is n*x^(n-1).
Step 3: Apply the power rule to each term in the function f(x).
Step 4: Differentiate the first term x^4. Using the power rule, the derivative is 4*x^(4-1) = 4x^3.
Step 5: Differentiate the second term -4x^3. Using the power rule, the derivative is -4*3*x^(3-1) = -12x^2.
Step 6: Differentiate the third term 6x^2. Using the power rule, the derivative is 6*2*x^(2-1) = 12x.
Step 7: Differentiate the constant term -2. The derivative of a constant is 0.
Step 8: Combine all the derivatives from steps 4, 5, 6, and 7 to get the final derivative: f'(x) = 4x^3 - 12x^2 + 12x + 0.
Step 9: Simplify the expression if necessary. In this case, it remains f'(x) = 4x^3 - 12x^2 + 12x.

Power Rule – The power rule states that the derivative of x^n is n*x^(n-1).
Polynomial Derivatives – Finding the derivative of polynomial functions involves applying the power rule to each term.