Answer:
[tex]\displaystyle \frac{d}{dx}[e^{2x}] = 2e^{2x}[/tex]
[tex]\displaystyle \frac{d}{dx}[e^{3x}] = 3e^{3x}[/tex]
General Formulas and Concepts:
Algebra I
- Terms/Coefficients
- Exponential Rule [Multiplying]: [tex]\displaystyle b^m \cdot b^n = b^{m + n}[/tex]
Calculus
Derivatives
Derivative Notation
eˣ Derivative: [tex]\displaystyle \frac{d}{dx}[e^x] = e^x[/tex]
Derivative Rule [Product Rule]: [tex]\displaystyle \frac{d}{dx} [f(x)g(x)]=f'(x)g(x) + g'(x)f(x)[/tex]
Step-by-step explanation:
Step 1: Define
[tex]\displaystyle \frac{d}{dx}[e^{2x}] = \frac{d}{dx}[e^x \cdot e^x][/tex]
[tex]\displaystyle \frac{d}{dx}[e^{3x}] = \frac{d}{dx}[e^x \cdot e^{2x}][/tex]
Step 2: Differentiate
[tex]\displaystyle \frac{d}{dx}[e^{2x}][/tex]
- [Derivative] Product Rule: [tex]\displaystyle \frac{d}{dx}[e^{2x}] = \frac{d}{dx}[e^x]e^x + e^x\frac{d}{dx}[e^x][/tex]
- [Derivative] eˣ Derivative: [tex]\displaystyle \frac{d}{dx}[e^{2x}] = e^x \cdot e^x + e^x \cdot e^x[/tex]
- [Derivative] Multiply [Exponential Rule - Multiplying]: [tex]\displaystyle \frac{d}{dx}[e^{2x}] = e^{2x} + e^{2x}[/tex]
- [Derivative] Combine like terms [Addition]: [tex]\displaystyle \frac{d}{dx}[e^{2x}] = 2e^{2x}[/tex]
[tex]\displaystyle \frac{d}{dx}[e^{3x}][/tex]
- [Derivative] Product Rule: [tex]\displaystyle \frac{d}{dx}[e^{3x}] = \frac{d}{dx}[e^x]e^{2x} + e^x\frac{d}{dx}[e^{2x}][/tex]
- [Derivative] eˣ Derivatives: [tex]\displaystyle \frac{d}{dx}[e^{3x}] = e^x(e^{2x}) + e^x(2e^{2x})[/tex]
- [Derivative] Multiply [Exponential Rule - Multiplying]: [tex]\displaystyle \frac{d}{dx}[e^{3x}] = e^{3x} + 2e^{3x}[/tex]
- [Derivative] Combine like terms [Addition]: [tex]\displaystyle \frac{d}{dx}[e^{3x}] = 3e^{3x}[/tex]
Topic: AP Calculus AB/BC (Calculus I/II)
Unit: Derivatives
Book: College Calculus 10e
