Question

Please someone help me with the question

Answer 1

Answer:

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

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

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

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

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

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

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

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

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

Unit: Derivatives

Book: College Calculus 10e