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Find the derivative of f(x) = x^3 - 3x^2 + 4 at x = 2.
Find the derivative of f(x) = x^3 - 3x^2 + 4 at x = 2.
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Practice Questions
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Q1
Find the derivative of f(x) = x^3 - 3x^2 + 4 at x = 2.
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12
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f'(x) = 3x^2 - 6x. At x = 2, f'(2) = 3(2^2) - 6(2) = 12 - 12 = 0.
Questions & Step-by-step Solutions
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Q
Q: Find the derivative of f(x) = x^3 - 3x^2 + 4 at x = 2.
Solution:
f'(x) = 3x^2 - 6x. At x = 2, f'(2) = 3(2^2) - 6(2) = 12 - 12 = 0.
Steps: 13
Show Steps
Step 1: Write down the function f(x) = x^3 - 3x^2 + 4.
Step 2: Find the derivative of f(x). The derivative f'(x) is found by applying the power rule: for x^n, the derivative is n*x^(n-1).
Step 3: Differentiate each term in f(x):
- The derivative of x^3 is 3x^2.
- The derivative of -3x^2 is -6x.
- The derivative of the constant 4 is 0.
Step 4: Combine the derivatives to get f'(x) = 3x^2 - 6x.
Step 5: Now, we need to find the value of the derivative at x = 2. Substitute 2 into f'(x).
Step 6: Calculate f'(2) = 3(2^2) - 6(2).
Step 7: Calculate 2^2, which is 4. Then multiply: 3 * 4 = 12.
Step 8: Calculate 6 * 2, which is 12.
Step 9: Now, subtract: 12 - 12 = 0.
Step 10: The derivative of f(x) at x = 2 is 0.
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