Answer:

General Formulas and Concepts:
Pre-Algebra
Order of Operations: BPEMDAS
- Brackets
- Parenthesis
- Exponents
- Multiplication
- Division
- Addition
- Subtraction
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)](https://img.qammunity.org/2022/formulas/mathematics/college/c6fshhoq1mws6w0d0la17c7k2dcytwd8kg.png)
Derivative Rule [Chain Rule]:
![\displaystyle (d)/(dx)[f(g(x))] =f'(g(x)) \cdot g'(x)](https://img.qammunity.org/2022/formulas/mathematics/high-school/vue68srn3fe6bds4idxorm97z7tgwelamw.png)
Explanation:
Step 1: Define
Identify
y = (3x - 1)⁵(4 - x⁴)⁵
Step 2: Differentiate
- Product Rule:
^5 + (3x - 1)^5(d)/(dx)[(4 - x^4)^5]](https://img.qammunity.org/2022/formulas/mathematics/college/16pdnb27uobcqrjmi1r0koe5508v1nceny.png)
- Chain Rule [Basic Power Rule]:
![\displaystyle y' =[5(3x - 1)^(5-1) \cdot (d)/(dx)[3x - 1]](4 - x^4)^5 + (3x - 1)^5[5(4 - x^4)^(5-1) \cdot (d)/(dx)[(4 - x^4)]]](https://img.qammunity.org/2022/formulas/mathematics/college/tg4ei73r58xlacezi302a5ej8dxihf7qdp.png)
- Simplify:
![\displaystyle y' =[5(3x - 1)^4 \cdot (d)/(dx)[3x - 1]](4 - x^4)^5 + (3x - 1)^5[5(4 - x^4)^4 \cdot (d)/(dx)[(4 - x^4)]]](https://img.qammunity.org/2022/formulas/mathematics/college/z242g7sqv3a7xoj35niq4f9c6t220wg8nw.png)
- Basic Power Rule:
^5 + (3x - 1)^5[5(4 - x^4)^4 \cdot -4x^(4-1)]](https://img.qammunity.org/2022/formulas/mathematics/college/t728iew1rvs4j3vizy1gvq7kvc8gtwpu3n.png)
- Simplify:
^5 + (3x - 1)^5[5(4 - x^4)^4 \cdot -4x^3]](https://img.qammunity.org/2022/formulas/mathematics/college/x5l583udgoqpov1e76ovhkuk2vqnb3f7l0.png)
- Multiply:

- Factor:
![\displaystyle y' = 5(3x-1)^4(4 - x^4)^4\bigg[ 3(4 - x^4) - 4x^3(3x - 1) \bigg]](https://img.qammunity.org/2022/formulas/mathematics/college/j1e2cwp2hmarhuino09dxtb988xeng86u1.png)
- [Distributive Property] Distribute 3:
![\displaystyle y' = 5(3x-1)^4(4 - x^4)^4\bigg[ 12 - 3x^4 - 4x^3(3x - 1) \bigg]](https://img.qammunity.org/2022/formulas/mathematics/college/7qw3vh5un6oz5vabd5kvh82s1ou6blzdba.png)
- [Distributive Property] Distribute -4x³:
![\displaystyle y' = 5(3x-1)^4(4 - x^4)^4\bigg[ 12 - 3x^4 - 12x^4 + 4x^3 \bigg]](https://img.qammunity.org/2022/formulas/mathematics/college/yg453duoa522ndy58w7lmx79bfgb1k10h6.png)
- [Brackets] Combine like terms:

- Factor:

Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Derivatives
Book: College Calculus 10e