Mathematics, 13.06.2021 21:20 delayne22

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

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Answer from: Isaacochoa780

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

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

Brackets Parenthesis Exponents Multiplication Division Addition Subtraction Left to Right

Distributive Property

Algebra I

Terms/CoefficientsFactoring

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]: $\displaystyle \frac{d}{dx} [f(x)g(x)]=f'(x)g(x) + g'(x)f(x)$

Derivative Rule [Chain Rule]: $\displaystyle \frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)$

Step-by-step explanation:

Step 1: Define

Identify

y = (2x - 5)²(5 - x⁵)²

Step 2: Differentiate

Derivative Rule [Product Rule]: $\displaystyle y' = \frac{d}{dx}[(2x - 5)^2](5 - x^5)^2 + (2x - 5)^2\frac{d}{dx}[(5 - x^5)^2]$ Chain Rule [Basic Power Rule]: $\displaystyle y' = [2(2x - 5)^{2-1} \cdot \frac{d}{dx}[2x]](5 - x^5)^2 + (2x - 5)^2[2(5 - x^5)^{2-1} \cdot \frac{d}{dx}[-x^5]]$ Simplify: $\displaystyle y' = [2(2x - 5) \cdot \frac{d}{dx}[2x]](5 - x^5)^2 + (2x - 5)^2[2(5 - x^5) \cdot \frac{d}{dx}[-x^5]]$ Basic Power Rule: $\displaystyle y' = [2(2x - 5) \cdot 1(2x^{1 - 1})](5 - x^5)^2 + (2x - 5)^2[2(5 - x^5) \cdot -5x^{5 - 1}]$ Simplify: $\displaystyle y' = [2(2x - 5) \cdot 2](5 - x^5)^2 + (2x - 5)^2[2(5 - x^5) \cdot -5x^4]$ Multiply: $\displaystyle y' = 4(2x - 5)(5 - x^5)^2 - 10x^4(2x - 5)^2(5 - x^5)$ Factor: $\displaystyle y' = 2(2x - 5)(5 - x^5)[2(5 - x^5) - 5x^4(2x - 5)]$ [Distributive Property] Distribute 2: $\displaystyle y' = 2(2x - 5)(5 - x^5)[10 - 2x^5 - 5x^4(2x - 5)]$ [Distributive Property] Distribute 5x⁴: $\displaystyle y' = 2(2x - 5)(5 - x^5)[10 - 2x^5 - 10x^5 + 25x^4]$ [Addition] Combine like terms (x⁵): $\displaystyle y' = 2(2x - 5)(5 - x^5)(10 - 12x^5 + 25x^4)$ 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

Answer from: Quest

which expression represents the approximate length of bc?

step-by-step explanation:

Answer from: Quest

step-by-step explanation:

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