Question

Differentiate the function.

y = (4x − 1)^2 (4 -x^5)^4

dy/dx=​

Answer 1

`\displaystyle y' = -4(4x - 1)(4 - x^5)^3(22x^5 - 5x^4 - 8)`

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

1. Brackets
2. Parenthesis
3. Exponents
4. Multiplication
5. Division
6. Addition
7. Subtraction

• Left to Right

Distributive Property

Algebra I

• Terms/Coefficients
• Factoring

Calculus

Derivatives

Derivative Notation

Derivative of a constant is 0

Basic Power Rule:

• f(x) = cxⁿ
• f’(x) = c·nxⁿ⁻¹

Derivative Rule [Product Rule]:
`\displaystyle (d)/(dx) [f(x)g(x)]=f'(x)g(x) + g'(x)f(x)`

Derivative Rule [Chain Rule]:
`\displaystyle (d)/(dx)[f(g(x))] =f'(g(x)) \cdot g'(x)`

Explanation:

Step 1: Define

Identify

y = (4x - 1)²(4 - x⁵)⁴

Step 2: Differentiate

1. Product Rule:
   `\displaystyle y' = (d)/(dx)[(4x - 1)^2](4 - x^5)^4 + (4x - 1)^2(d)/(dx)[(4 - x^5)^4]`
2. Chain Rule [Basic Power Rule]:
   `\displaystyle y' = [2(4x - 1)^(2 - 1) \cdot (d)/(dx)[(4x - 1)]](4 - x^5)^4 + (4x - 1)^2[4(4 - x^5)^(4 - 1) \cdot (d)/(dx)[(4 - x^5)]]`
3. Simplify:
   `\displaystyle y' = [2(4x - 1) \cdot (d)/(dx)[(4x - 1)]](4 - x^5)^4 + (4x - 1)^2[4(4 - x^5)^3 \cdot (d)/(dx)[(4 - x^5)]]`
4. Basic Power Rule:
   `\displaystyle y' = [2(4x - 1) \cdot 4x^(1 - 1)](4 - x^5)^4 + (4x - 1)^2[4(4 - x^5)^3 \cdot -5x^(5 - 1)]`
5. Simplify:
   `\displaystyle y' = [2(4x - 1) \cdot 4](4 - x^5)^4 + (4x - 1)^2[4(4 - x^5)^3 \cdot -5x^4]`
6. Multiply:
   `\displaystyle y' = 8(4x - 1)(4 - x^5)^4 - 20x^4(4x - 1)^2(4 - x^5)^3`
7. Factor:
   `\displaystyle y' = 4(4x - 1)(4 - x^5)^3 \bigg[ 2(4 - x^5) - 5x^4(4x - 1) \bigg]`
8. [Distributive Property] Distribute 2:
   `\displaystyle y' = 4(4x - 1)(4 - x^5)^3 \bigg[ 8 - 2x^5 - 5x^4(4x - 1) \bigg]`
9. [Distributive Property] Distribute -5x⁴:
   `\displaystyle y' = 4(4x - 1)(4 - x^5)^3 \bigg[ 8 - 2x^5 - 20x^5 + 5x^4 \bigg]`
10. [Brackets] Combine like terms:
    `\displaystyle y' = 4(4x - 1)(4 - x^5)^3(-22x^5 + 5x^4 + 8)`
11. Factor:
    `\displaystyle y' = -4(4x - 1)(4 - x^5)^3(22x^5 - 5x^4 - 8)`

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

Unit: Derivatives

Book: College Calculus 10e