Question

Differentiate the function, y = (2x - 5)^2 (5-x^5)^2?​

Answer 1

`\displaystyle y' = 2(2x - 5)(x^5 - 5)(12x^5 - 25x^4 - 10)`

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 = (2x - 5)²(5 - x⁵)²

Step 2: Differentiate

1. Derivative Rule [Product Rule]:
   `\displaystyle y' = (d)/(dx)[(2x - 5)^2](5 - x^5)^2 + (2x - 5)^2(d)/(dx)[(5 - x^5)^2]`
2. Chain Rule [Basic Power Rule]:
   `\displaystyle y' = [2(2x - 5)^(2-1) \cdot (d)/(dx)[2x]](5 - x^5)^2 + (2x - 5)^2[2(5 - x^5)^(2-1) \cdot (d)/(dx)[-x^5]]`
3. Simplify:
   `\displaystyle y' = [2(2x - 5) \cdot (d)/(dx)[2x]](5 - x^5)^2 + (2x - 5)^2[2(5 - x^5) \cdot (d)/(dx)[-x^5]]`
4. Basic Power Rule:
   `\displaystyle y' = [2(2x - 5) \cdot 1(2x^(1 - 1))](5 - x^5)^2 + (2x - 5)^2[2(5 - x^5) \cdot -5x^(5 - 1)]`
5. Simplify:
   `\displaystyle y' = [2(2x - 5) \cdot 2](5 - x^5)^2 + (2x - 5)^2[2(5 - x^5) \cdot -5x^4]`
6. Multiply:
   `\displaystyle y' = 4(2x - 5)(5 - x^5)^2 - 10x^4(2x - 5)^2(5 - x^5)`
7. Factor:
   `\displaystyle y' = 2(2x - 5)(5 - x^5)[2(5 - x^5) - 5x^4(2x - 5)]`
8. [Distributive Property] Distribute 2:
   `\displaystyle y' = 2(2x - 5)(5 - x^5)[10 - 2x^5 - 5x^4(2x - 5)]`
9. [Distributive Property] Distribute 5x⁴:
   `\displaystyle y' = 2(2x - 5)(5 - x^5)[10 - 2x^5 - 10x^5 + 25x^4]`
10. [Addition] Combine like terms (x⁵):
    `\displaystyle y' = 2(2x - 5)(5 - x^5)(10 - 12x^5 + 25x^4)`
11. Rewrite:
    `\displaystyle y' = 2(2x - 5)(x^5 - 5)(12x^5 - 25x^4 - 10)`

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

Unit: Derivatives

Book: College Calculus 10e