Question

Simple true or false... d/dx{x^3e^x]=x^3e^x(3x+2) True or false

Answer 1

Answer:

False.

General Formulas and Concepts:

Algebra I

Terms/Coefficients

• Factoring

Calculus

Differentiation

• Derivatives
• Derivative Notation

Derivative Rule [Basic Power Rule]:

1. f(x) = cxⁿ
2. f’(x) = c·nxⁿ⁻¹

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

Identify.

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

Step 2: Differentiate

1. Derivative Rule [Product Rule]:                                                                     $\displaystyle \frac{d}{dx}[x^3e^x] = \frac{d}{dx}[x^3]e^x + x^3 \frac{d}{dx}[e^x]$
2. Derivative Rule [Basic Power Rule]:                                                             $\displaystyle \frac{d}{dx}[x^3e^x] = 3x^2e^x + x^3 \frac{d}{dx}[e^x]$
3. Exponential Differentiation:                                                                         $\displaystyle \frac{d}{dx}[x^3e^x] = 3x^2e^x + x^3e^x$
4. Factor:                                                                                                           $\displaystyle \frac{d}{dx}[x^3e^x] = (x^3 + 3x^2)e^x$

∴ $\displaystyle \frac{d}{dx}[x^3e^x] \neq x^3e^x(3x + 2)$  but  $\displaystyle \frac{d}{dx}[x^3e^x] = (x^3 + 3x^2)e^x$.

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

Unit: Differentiation