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Answer:

$\displaystyle \frac{d}{dx}[e^{2x}] = 2e^{2x}$

$\displaystyle \frac{d}{dx}[e^{3x}] = 3e^{3x}$

General Formulas and Concepts:

Algebra I

Calculus

Derivatives

Derivative Notation

eˣ Derivative: $\displaystyle \frac{d}{dx}[e^x] = e^x$

Derivative Rule [Product Rule]: $\displaystyle \frac{d}{dx} [f(x)g(x)]=f'(x)g(x) + g'(x)f(x)$

Step-by-step explanation:

Step 1: Define

$\displaystyle \frac{d}{dx}[e^{2x}] = \frac{d}{dx}[e^x \cdot e^x]$

$\displaystyle \frac{d}{dx}[e^{3x}] = \frac{d}{dx}[e^x \cdot e^{2x}]$

Step 2: Differentiate

$\displaystyle \frac{d}{dx}[e^{2x}]$

[Derivative] Product Rule: $\displaystyle \frac{d}{dx}[e^{2x}] = \frac{d}{dx}[e^x]e^x + e^x\frac{d}{dx}[e^x]$
[Derivative] eˣ Derivative: $\displaystyle \frac{d}{dx}[e^{2x}] = e^x \cdot e^x + e^x \cdot e^x$
[Derivative] Multiply [Exponential Rule - Multiplying]: $\displaystyle \frac{d}{dx}[e^{2x}] = e^{2x} + e^{2x}$
[Derivative] Combine like terms [Addition]: $\displaystyle \frac{d}{dx}[e^{2x}] = 2e^{2x}$

$\displaystyle \frac{d}{dx}[e^{3x}]$

[Derivative] Product Rule: $\displaystyle \frac{d}{dx}[e^{3x}] = \frac{d}{dx}[e^x]e^{2x} + e^x\frac{d}{dx}[e^{2x}]$
[Derivative] eˣ Derivatives: $\displaystyle \frac{d}{dx}[e^{3x}] = e^x(e^{2x}) + e^x(2e^{2x})$
[Derivative] Multiply [Exponential Rule - Multiplying]: $\displaystyle \frac{d}{dx}[e^{3x}] = e^{3x} + 2e^{3x}$
[Derivative] Combine like terms [Addition]: $\displaystyle \frac{d}{dx}[e^{3x}] = 3e^{3x}$