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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.
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Practice Questions
1 question
Q1
Find the derivative of f(x) = e^x * ln(x) at x = 1.
1
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e
ln(e)
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Using the product rule, f'(x) = e^x * ln(x) + e^x/x. At x = 1, this simplifies to 0.
Questions & Step-by-step Solutions
1 item
Q
Q: Find the derivative of f(x) = e^x * ln(x) at x = 1.
Solution:
Using the product rule, f'(x) = e^x * ln(x) + e^x/x. At x = 1, this simplifies to 0.
Steps: 10
Show Steps
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.
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