Find the derivative of f(x) = x^3 - 3x^2 + 4x - 5.

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Question: Find the derivative of f(x) = x^3 - 3x^2 + 4x - 5.

Options:

Correct Answer: 3x^2 - 6x + 4

Solution:

Using the power rule, f\'(x) = 3x^2 - 6x + 4.

Find the derivative of f(x) = x^3 - 3x^2 + 4x - 5.

Find the derivative of f(x) = x^3 - 3x^2 + 4x - 5.

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

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.