If f(x) = 5x^2 + 3x - 1, what is f''(x)? (2020)

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Question: If f(x) = 5x^2 + 3x - 1, what is f\'\'(x)? (2020)

Options:

Correct Answer: 10

Exam Year: 2020

Solution:

The first derivative f\'(x) = 10x + 3, and the second derivative f\'\'(x) = 10.

If f(x) = 5x^2 + 3x - 1, what is f''(x)? (2020)

If f(x) = 5x^2 + 3x - 1, what is f''(x)? (2020)

Step 1: Start with the function f(x) = 5x^2 + 3x - 1.
Step 2: Find the first derivative f'(x) by using the power rule. The power rule states that if f(x) = ax^n, then f'(x) = n * ax^(n-1).
Step 3: Apply the power rule to each term in f(x):
- For the term 5x^2, the derivative is 2 * 5 * x^(2-1) = 10x.
- For the term 3x, the derivative is 1 * 3 * x^(1-1) = 3.
- The constant term -1 has a derivative of 0.
Step 4: Combine the derivatives from each term to get f'(x) = 10x + 3.
Step 5: Now, find the second derivative f''(x) by differentiating f'(x).
Step 6: Again, apply the power rule to f'(x):
- For the term 10x, the derivative is 1 * 10 * x^(1-1) = 10.
- The constant term 3 has a derivative of 0.
Step 7: Combine the derivatives to get f''(x) = 10.

Differentiation – The process of finding the derivative of a function, which measures the rate of change.
Second Derivative – The derivative of the first derivative, which provides information about the curvature of the function.