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

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

Options:

Correct Answer: 3x^2 - 8x + 6

Solution:

Using the power rule, f\'(x) = 3x^2 - 8x + 6.

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

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

Correct Answer: f'(x) = 3x^2 - 8x + 6

Step 1: Identify the function you want to differentiate, which is f(x) = x^3 - 4x^2 + 6x.
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: For the first term x^3, the derivative is 3*x^(3-1) = 3x^2.
Step 5: For the second term -4x^2, the derivative is -4*2*x^(2-1) = -8x.
Step 6: For the third term 6x, the derivative is 6*1*x^(1-1) = 6.
Step 7: Combine all the derivatives from the previous steps to get the final derivative: f'(x) = 3x^2 - 8x + 6.

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.