Math Problem Statement
if f(x) = e^(3x) *cos(2x), than f'(x)=
Solution
To find the derivative of , we will use the product rule and chain rule.
The product rule states: Here:
Step 1: Differentiate
Using the chain rule:
Step 2: Differentiate
Again using the chain rule:
Step 3: Apply the product rule
Now applying the product rule:
Final Answer:
Would you like further explanation on any steps or a breakdown of the product and chain rules?
Here are five related questions to deepen your understanding:
- How would you apply the product rule to ?
- What is the derivative of ?
- How would the second derivative be computed for this function?
- What is the chain rule and how does it apply to exponential functions like ?
- How do you handle the differentiation of trigonometric functions like or ?
Tip: When dealing with derivatives involving products of functions, always check if the product or chain rule applies, as they are crucial for handling such cases efficiently.
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Math Problem Analysis
Mathematical Concepts
Calculus
Differentiation
Product Rule
Chain Rule
Exponential Functions
Trigonometric Functions
Formulas
Product Rule: (f ⋅ g)' = f' ⋅ g + f ⋅ g'
Chain Rule: d/dx[f(g(x))] = f'(g(x)) ⋅ g'(x)
Derivative of e^(kx): d/dx[e^(kx)] = k ⋅ e^(kx)
Derivative of cos(ax): d/dx[cos(ax)] = -a ⋅ sin(ax)
Theorems
Product Rule
Chain Rule
Suitable Grade Level
Advanced High School (Grades 11-12), College Level