Mathematics, 13.06.2021 21:10 alexj29227405

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

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Answer from: bree2837

$\displaystyle y' = -(2x - 5)(6x - 25)$

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 - x)

Step 2: Differentiate

Derivative Rule [Product Rule]: $\displaystyle y' = \frac{d}{dx}[(2x - 5)^2](5 - x) + (2x - 5)^2\frac{d}{dx}[(5 - x)]$ Chain Rule [Basic Power Rule]: $\displaystyle y' = [2(2x - 5)^{2 - 1} \cdot \frac{d}{dx}[2x]](5 - x) + (2x - 5)^2\frac{d}{dx}[(5 - x)]$ Simplify: $\displaystyle y' = [2(2x - 5) \cdot \frac{d}{dx}[2x]](5 - x) + (2x - 5)^2\frac{d}{dx}[(5 - x)]$ Basic Power Rule: $\displaystyle y' = [2(2x - 5) \cdot 1 \cdot 2x^{1 - 1}](5 - x) + (2x - 5)^2(1 \cdot -x^{1 - 1})]$ Simplify: $\displaystyle y' = [2(2x - 5) \cdot 2](5 - x) + (2x - 5)^2(-1)$ Multiply: $\displaystyle y' = 4(2x - 5)(5 - x) - (2x - 5)^2$ Factor: $\displaystyle y' = (2x - 5)[4(5 - x) - (2x - 5)]$ [Distributive Property] Distribute 4: $\displaystyle y' = (2x - 5)[20 - 4x - (2x - 5)]$ [Distributive Property] Distribute negative: $\displaystyle y' = (2x - 5)[20 - 4x - 2x + 5]$ [Subtraction] Combine like terms (x): $\displaystyle y' = (2x - 5)[20 - 6x + 5]$ [Addition] Combine like terms: $\displaystyle y' = (2x - 5)(25 - 6x)$ Factor: $\displaystyle y' = -(2x - 5)(6x - 25)$

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

Unit: Derivatives

Book: College Calculus 10e

Answer from: Quest

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Answer from: Quest

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