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Find the second derivative of f(x) = e^x at x = 0.

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Question: Find the second derivative of f(x) = e^x at x = 0.

Options:

  1. 0
  2. 1
  3. e
  4. e^2

Correct Answer: 1

Solution: 

f\'\'(x) = e^x, thus f\'\'(0) = e^0 = 1.

Find the second derivative of f(x) = e^x at x = 0.

Practice Questions

Q1
Find the second derivative of f(x) = e^x at x = 0.
  1. 0
  2. 1
  3. e
  4. e^2

Questions & Step-by-Step Solutions

Find the second derivative of f(x) = e^x at x = 0.
  • Step 1: Identify the function we need to work with, which is f(x) = e^x.
  • Step 2: Find the first derivative of f(x). The derivative of e^x is e^x, so f'(x) = e^x.
  • Step 3: Now, find the second derivative. The derivative of f'(x) = e^x is also e^x, so f''(x) = e^x.
  • Step 4: Evaluate the second derivative at x = 0. Substitute 0 into f''(x): f''(0) = e^0.
  • Step 5: Calculate e^0. Since e^0 = 1, we find that f''(0) = 1.
  • Differentiation – Understanding how to find the first and second derivatives of a function.
  • Exponential Functions – Recognizing the properties of the exponential function, particularly that its derivative is itself.
  • Evaluation at a Point – Evaluating the second derivative at a specific point, in this case, x = 0.
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