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

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

Options:

Correct Answer: 1

Solution:

Using the product rule, f\'(0) = e^0 * sin(0) + e^0 * cos(0) = 0 + 1 = 1.

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

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

Step 1: Identify the function f(x) = e^x * sin(x).
Step 2: Recognize that we need to find the derivative f'(x) using the product rule.
Step 3: Recall the product rule: if you have two functions u(x) and v(x), then the derivative is u'v + uv'.
Step 4: In our case, let u(x) = e^x and v(x) = sin(x).
Step 5: Find the derivatives: u'(x) = e^x and v'(x) = cos(x).
Step 6: Apply the product rule: f'(x) = u'v + uv' = e^x * sin(x) + e^x * cos(x).
Step 7: Now, we need to evaluate f'(x) at x = 0.
Step 8: Substitute x = 0 into f'(x): f'(0) = e^0 * sin(0) + e^0 * cos(0).
Step 9: Calculate e^0, which is 1, and sin(0), which is 0.
Step 10: Calculate cos(0), which is 1.
Step 11: Substitute these values into the equation: f'(0) = 1 * 0 + 1 * 1.
Step 12: Simplify the expression: f'(0) = 0 + 1 = 1.

Product Rule – The product rule is used to find the derivative of a product of two functions, stating that (uv)' = u'v + uv'.
Exponential and Trigonometric Functions – Understanding the derivatives of e^x and trigonometric functions like sin(x) and cos(x) is essential.
Evaluating Derivatives at a Point – Finding the value of the derivative at a specific point, in this case, x = 0.