What is the second derivative of f(x) = e^x sin(x)?

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Question: What is the second derivative of f(x) = e^x sin(x)?

Options:

Correct Answer: e^x (sin(x) + cos(x))

Solution:

Using the product rule and chain rule, the second derivative is f\'\'(x) = e^x (sin(x) + cos(x)).

What is the second derivative of f(x) = e^x sin(x)?

What is the second derivative of f(x) = e^x sin(x)?

Step 1: Identify the function f(x) = e^x sin(x).
Step 2: To find the second derivative, we first need to find the first derivative f'(x).
Step 3: Use the product rule for differentiation, which states that if you have two functions u(x) and v(x), then the derivative is u'v + uv'.
Step 4: In our case, let u = e^x and v = sin(x).
Step 5: Calculate the derivatives: u' = e^x and v' = cos(x).
Step 6: Apply the product rule: f'(x) = u'v + uv' = e^x sin(x) + e^x cos(x).
Step 7: Simplify the first derivative: f'(x) = e^x (sin(x) + cos(x)).
Step 8: Now, we need to find the second derivative f''(x).
Step 9: Differentiate f'(x) = e^x (sin(x) + cos(x)) again using the product rule.
Step 10: Let u = e^x and v = (sin(x) + cos(x)).
Step 11: Calculate the derivatives: u' = e^x and v' = cos(x) - sin(x).
Step 12: Apply the product rule again: f''(x) = u'v + uv' = e^x (sin(x) + cos(x)) + e^x (cos(x) - sin(x)).
Step 13: Combine like terms: f''(x) = e^x (sin(x) + cos(x) + cos(x) - sin(x)).
Step 14: Simplify: f''(x) = e^x (2cos(x)).

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