Mathematics, 19.11.2020 14:00 brenton14

Product and quotient rule ,differentiate each function with respect to x, in calculus y = (-4x^5+4) * 5x^⁵

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Answer from: chancegodwin5

$\displaystyle y' = -100x^4(2x^5 - 1)$

General Formulas and Concepts:

Pre-Algebra

Distributive Property

Algebra I

Terms/CoefficientsFactoringFunctionsFunction Notation

Calculus

Derivatives

Derivative Notation

Derivative Property [Multiplied Constant]: $\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)$

Derivative Property [Addition/Subtraction]: $\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)]$

Basic Power Rule:

f(x) = cxⁿ f’(x) = c·nxⁿ⁻¹

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

Step-by-step explanation:

Step 1: Define

Identify

$\displaystyle y = (-4x^5 + 4)5x^5$

Step 2: Differentiate

Product Rule: $\displaystyle y' = \frac{d}{dx}[(-4x^5 + 4)]5x^5 + (-4x^5 + 4)\frac{d}{dx}[5x^5]$ Basic Power Rule [Derivative Property - Addition/Subtraction]: $\displaystyle y' = (5 \cdot -4x^{5 - 1} + 0)5x^5 + (-4x^5 + 4)(5 \cdot 5x^{5 - 1})$ Simplify: $\displaystyle y' = (-20x^4)5x^5 + (-4x^5 + 4)(25x^4)$ Factor: $\displaystyle y' = 5x^4 \bigg[ (-20x^4)x + (-4x^5 + 4)5 \bigg]$ [Distributive Property] Distributive parenthesis: $\displaystyle y' = 5x^4 \bigg[ -20x^5 - 20x^5 + 20 \bigg]$ Combine like terms: $\displaystyle y' = 5x^4(-40x^5 + 20)$ Factor: $\displaystyle y' = -100x^4(2x^5 - 1)$

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

Unit: Derivatives

Book: College Calculus 10e

Answer from: Quest

yess

step-by-step explanation:

idk whyy this is happening, i cant even read my notification bc i keep getting errors

~coco

Answer from: Quest

For range i think it’s area

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Product and quotient rule ,differentiate each function with respect to x, in calculus y = (-4x^5+4)...

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