Question

Please someone help me with the question

Answer 1

`\displaystyle (d)/(dx)[e^(2x)] = 2e^(2x)`

`\displaystyle (d)/(dx)[e^(3x)] = 3e^(3x)`

General Formulas and Concepts:

Algebra I

• Terms/Coefficients
• Exponential Rule [Multiplying]:
  `\displaystyle b^m \cdot b^n = b^(m + n)`

Calculus

Derivatives

Derivative Notation

eˣ Derivative:
`\displaystyle (d)/(dx)[e^x] = e^x`

Derivative Rule [Product Rule]:
`\displaystyle (d)/(dx) [f(x)g(x)]=f'(x)g(x) + g'(x)f(x)`

Explanation:

Step 1: Define

`\displaystyle (d)/(dx)[e^(2x)] = (d)/(dx)[e^x \cdot e^x]`

`\displaystyle (d)/(dx)[e^(3x)] = (d)/(dx)[e^x \cdot e^(2x)]`

Step 2: Differentiate

`\displaystyle (d)/(dx)[e^(2x)]`

1. [Derivative] Product Rule:
   `\displaystyle (d)/(dx)[e^(2x)] = (d)/(dx)[e^x]e^x + e^x(d)/(dx)[e^x]`
2. [Derivative] eˣ Derivative:
   `\displaystyle (d)/(dx)[e^(2x)] = e^x \cdot e^x + e^x \cdot e^x`
3. [Derivative] Multiply [Exponential Rule - Multiplying]:
   `\displaystyle (d)/(dx)[e^(2x)] = e^(2x) + e^(2x)`
4. [Derivative] Combine like terms [Addition]:
   `\displaystyle (d)/(dx)[e^(2x)] = 2e^(2x)`

`\displaystyle (d)/(dx)[e^(3x)]`

1. [Derivative] Product Rule:
   `\displaystyle (d)/(dx)[e^(3x)] = (d)/(dx)[e^x]e^(2x) + e^x(d)/(dx)[e^(2x)]`
2. [Derivative] eˣ Derivatives:
   `\displaystyle (d)/(dx)[e^(3x)] = e^x(e^(2x)) + e^x(2e^(2x))`
3. [Derivative] Multiply [Exponential Rule - Multiplying]:
   `\displaystyle (d)/(dx)[e^(3x)] = e^(3x) + 2e^(3x)`
4. [Derivative] Combine like terms [Addition]:
   `\displaystyle (d)/(dx)[e^(3x)] = 3e^(3x)`

Topic: AP Calculus AB/BC (Calculus I/II)

Unit: Derivatives

Book: College Calculus 10e