Question

Product and quotient rule ,differentiate each function with respect to x, in calculus

y = (-4x^5+4) * 5x^⁵

Answer 1

`\displaystyle y' = -100x^4(2x^5 - 1)`

General Formulas and Concepts:

Pre-Algebra

• Distributive Property

Algebra I

• Terms/Coefficients
• Factoring
• Functions
• Function Notation

Calculus

Derivatives

Derivative Notation

Derivative Property [Multiplied Constant]:
`\displaystyle (d)/(dx) [cf(x)] = c \cdot f'(x)`

Derivative Property [Addition/Subtraction]:
`\displaystyle (d)/(dx)[f(x) + g(x)] = (d)/(dx)[f(x)] + (d)/(dx)[g(x)]`

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 y = (-4x^5 + 4)5x^5`

Step 2: Differentiate

1. Product Rule:
   `\displaystyle y' = (d)/(dx)[(-4x^5 + 4)]5x^5 + (-4x^5 + 4)(d)/(dx)[5x^5]`
2. Basic Power Rule [Derivative Property - Addition/Subtraction]:
   `\displaystyle y' = (5 \cdot -4x^(5 - 1) + 0)5x^5 + (-4x^5 + 4)(5 \cdot 5x^(5 - 1))`
3. Simplify:
   `\displaystyle y' = (-20x^4)5x^5 + (-4x^5 + 4)(25x^4)`
4. Factor:
   `\displaystyle y' = 5x^4 \bigg[ (-20x^4)x + (-4x^5 + 4)5 \bigg]`
5. [Distributive Property] Distributive parenthesis:
   `\displaystyle y' = 5x^4 \bigg[ -20x^5 - 20x^5 + 20 \bigg]`
6. Combine like terms:
   `\displaystyle y' = 5x^4(-40x^5 + 20)`
7. Factor:
   `\displaystyle y' = -100x^4(2x^5 - 1)`

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

Unit: Derivatives

Book: College Calculus 10e