Find the derivative of f(x) = e^x * ln(x) at x = 1.

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Question: Find the derivative of f(x) = e^x * ln(x) at x = 1.

Options:

Correct Answer: 1

Solution:

Using the product rule, f\'(x) = e^x * ln(x) + e^x/x. At x = 1, this simplifies to 0.

Find the derivative of f(x) = e^x * ln(x) at x = 1.

Find the derivative of f(x) = e^x * ln(x) at x = 1.

Step 1: Identify the function f(x) = e^x * ln(x).
Step 2: Recognize that this is a product of two functions: u = e^x and v = ln(x).
Step 3: Recall the product rule for derivatives: (u * v)' = u' * v + u * v'.
Step 4: Calculate the derivative of u: u' = e^x (the derivative of e^x).
Step 5: Calculate the derivative of v: v' = 1/x (the derivative of ln(x)).
Step 6: Apply the product rule: f'(x) = u' * v + u * v' = e^x * ln(x) + e^x * (1/x).
Step 7: Simplify the expression: f'(x) = e^x * ln(x) + e^x/x.
Step 8: Substitute x = 1 into the derivative: f'(1) = e^1 * ln(1) + e^1/1.
Step 9: Calculate ln(1) = 0, so f'(1) = e * 0 + e = e.
Step 10: Since e is a constant, the final answer is f'(1) = e.

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 Logarithmic Functions – Understanding the properties and derivatives of exponential functions (e^x) and logarithmic functions (ln(x)).
Evaluation of Derivatives – Evaluating the derivative at a specific point, in this case, x = 1.