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Differentiate the function.

y = (4x − 1)^2 (4 -x^5)^4

dy/dx=​

User Cara
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Answer:


\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

User SamwellTarly
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