Math Problem Statement
Solution
We are asked to find the derivative of the function .
Step-by-Step Solution:
The function can be rewritten as:
We'll apply the product rule for differentiation, which states: Here, and .
Step 1: Differentiate
The derivative of is:
Step 2: Differentiate
The derivative of is:
Step 3: Apply the product rule
Now, apply the product rule: Simplifying both terms:
Step 4: Combine terms
Factor out :
Now, express everything under a common denominator:
This matches the option:
This corresponds to the second option in the image.
Would you like further clarification or any additional details on this problem?
Here are some related questions to explore:
- What is the product rule in calculus, and when do we use it?
- How do we differentiate exponential functions involving constants?
- What is the chain rule, and how would it apply if the exponent in were more complex?
- How can we simplify expressions involving radicals and exponents for derivatives?
- Why is factoring out common terms, like , helpful in derivative expressions?
Tip: When applying the product rule, always simplify intermediate results as much as possible before moving on to the next step!
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Math Problem Analysis
Mathematical Concepts
Product Rule
Exponential Functions
Differentiation
Formulas
Product Rule: (f * g)' = f' * g + f * g'
Power Rule: d/dx(x^n) = n * x^(n-1)
Exponential Derivative: d/dx(e^(kx)) = k * e^(kx)
Theorems
Product Rule
Power Rule
Exponential Function Derivative
Suitable Grade Level
Grades 11-12 / Early University Calculus
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