Question

Simple true or false...

d/dx{x^3e^x]=x^3e^x(3x+2)

True or false

Answer 1

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 (d)/(dx) [f(x)g(x)]=f'(x)g(x) + g'(x)f(x)`

Explanation:

Step 1: Define

Identify.

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

Step 2: Differentiate

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

∴
`\displaystyle (d)/(dx)[x^3e^x] \\eq x^3e^x(3x + 2)` but
`\displaystyle (d)/(dx)[x^3e^x] = (x^3 + 3x^2)e^x`.

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

Unit: Differentiation