Question: What is the derivative of f(x) = e^x * ln(x)? (2022)
Options:
e^x * ln(x)
e^x/x
e^x * (1 + ln(x))
e^x * ln(x)/x
Correct Answer: e^x * (1 + ln(x))
Exam Year: 2022
Solution:
Using the product rule, f\'(x) = e^x * ln(x) + e^x * (1/x) = e^x * (ln(x) + 1/x).
What is the derivative of f(x) = e^x * ln(x)? (2022)
Practice Questions
Q1
What is the derivative of f(x) = e^x * ln(x)? (2022)
e^x * ln(x)
e^x/x
e^x * (1 + ln(x))
e^x * ln(x)/x
Questions & Step-by-Step Solutions
What is the derivative of f(x) = e^x * ln(x)? (2022)
Step 1: Identify the function f(x) = e^x * ln(x). This is a product of two functions: e^x and ln(x).
Step 2: Recall the product rule for derivatives. The product rule states that if you have two functions u(x) and v(x), then the derivative of their product is u'v + uv'.
Step 3: Assign u = e^x and v = ln(x). Now we need to find the derivatives of u and v.
Step 4: Calculate the derivative of u: u' = d/dx(e^x) = e^x.
Step 5: Calculate the derivative of v: v' = d/dx(ln(x)) = 1/x.
