If f(x) = x^2 + 2x + 1, what is f''(x)? (2023)

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Question: If f(x) = x^2 + 2x + 1, what is f\'\'(x)? (2023)

Options:

Correct Answer: 2

Exam Year: 2023

Solution:

First derivative f\'(x) = 2x + 2. Second derivative f\'\'(x) = 2.

If f(x) = x^2 + 2x + 1, what is f''(x)? (2023)

If f(x) = x^2 + 2x + 1, what is f''(x)? (2023)

Step 1: Start with the function f(x) = x^2 + 2x + 1.
Step 2: Find the first derivative f'(x). To do this, use the power rule: the derivative of x^n is n*x^(n-1).
Step 3: Apply the power rule to each term in f(x):
- The derivative of x^2 is 2x.
- The derivative of 2x is 2.
- The derivative of 1 (a constant) is 0.
Step 4: Combine these results to get the first derivative: f'(x) = 2x + 2.
Step 5: Now, find the second derivative f''(x) by taking the derivative of f'(x).
Step 6: Again, apply the power rule to f'(x):
- The derivative of 2x is 2.
- The derivative of 2 (a constant) is 0.
Step 7: Combine these results to get the second derivative: f''(x) = 2.

Differentiation – The process of finding the derivative of a function, which measures how the function changes as its input changes.
Second Derivative – The derivative of the first derivative, which provides information about the curvature or concavity of the original function.