Math Problem Statement

if f(x) = e^(3x) *cos(2x), than f'(x)=

Solution

To find the derivative of f(x)=e3xcos(2x)f(x) = e^{3x} \cdot \cos(2x), we will use the product rule and chain rule.

The product rule states: (fg)=fg+fg(f \cdot g)' = f' \cdot g + f \cdot g' Here:

  • f(x)=e3xf(x) = e^{3x}
  • g(x)=cos(2x)g(x) = \cos(2x)

Step 1: Differentiate e3xe^{3x}

Using the chain rule: f(x)=ddx(e3x)=3e3xf'(x) = \frac{d}{dx}(e^{3x}) = 3e^{3x}

Step 2: Differentiate cos(2x)\cos(2x)

Again using the chain rule: g(x)=ddx(cos(2x))=2sin(2x)g'(x) = \frac{d}{dx}(\cos(2x)) = -2\sin(2x)

Step 3: Apply the product rule

Now applying the product rule: f(x)=(3e3x)cos(2x)+e3x(2sin(2x))f'(x) = (3e^{3x}) \cdot \cos(2x) + e^{3x} \cdot (-2\sin(2x))

Final Answer:

f(x)=3e3xcos(2x)2e3xsin(2x)f'(x) = 3e^{3x} \cos(2x) - 2e^{3x} \sin(2x)

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:

  1. How would you apply the product rule to f(x)=x2exf(x) = x^2 \cdot e^{x}?
  2. What is the derivative of sin(3x)ex\sin(3x) \cdot e^{x}?
  3. How would the second derivative f(x)f''(x) be computed for this function?
  4. What is the chain rule and how does it apply to exponential functions like ekxe^{kx}?
  5. How do you handle the differentiation of trigonometric functions like cos(ax)\cos(ax) or sin(bx)\sin(bx)?

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