Mathematics, 28.11.2020 01:00 issagirl05

Can someone help me with this please

Answers: 2

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Answer from: ReeseMoffitt8032

y'' = 18x(x³ + 8)⁴(17x³ + 16)

General Formulas and Concepts:

Calculus

Derivative Notation dy/dx

Derivative of a constant is 0.

Basic Power Rule:

f(x) = cxⁿ f’(x) = c·nxⁿ⁻¹

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

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

Step-by-step explanation:

Step 1: Define

y = (x³ + 8)⁶

Step 2: Find 1st Derivative (dy/dx)

Chain Rule [Basic Power]: y' = 6(x³ + 8)⁶⁻¹ · (3x³⁻¹ + 0)Simplify: y' = 6(x³ + 8)⁵ · 3x²Multiply: y' = 18x²(x³ + 8)⁵

Step 3: Find 2nd Derivative (d²y/dx²)

Product Rule [Chain/Basic Power]: y'' = 2 · 18x²⁻¹ · (x³ + 8)⁵ + 18x² · 5(x³ + 8)⁵⁻¹ · (3x³⁻¹ + 0)Simplify: y'' = 36x(x³ + 8)⁵ + 90x²(x³ + 8)⁴ · 3x²Multiply: y'' = 36x(x³ + 8)⁵ + 270x⁴(x³ + 8)⁴Factor: y'' = 6x(x³ + 8)⁴[6(x³ + 8) + 45x³]Distribute: y'' = 6x(x³ + 8)⁴[6x³ + 48 + 45x³]Combine like terms: y'' = 6x(x³ + 8)⁴[51x³ + 48]Factor: y'' = 6x(x³ + 8)⁴ · 3(17x³ + 16)Multiply: y'' = 18x(x³ + 8)⁴(17x³ + 16)

Answer from: montgomerykarloxc24x

$\frac{d^{2}y}{dx^{2}} = 180x.(x^{3} +8)^{4}$

Step-by-step explanation:

This question requires use of the chain rule:

$y = (x^{3} + 8)^{6}$

Let:

$u = x^{3} + 8$

Then:

$y = u^{6}$

And:

$\frac{dy}{dx} = \frac{dy}{du}. \frac{du}{dx}\\\\\frac{d^{2}y}{dx^{2} } = \frac{d^{2}y}{du^{2}}. \frac{d^{2}u}{dx^{2}}\\\\\frac{dy}{du} = 6u^{5}\\\\\frac{d^{2}y}{du^{2}} = 30u^{4}\\\\\frac{du}{dx} = 3x^{2}\\\\\frac{d^{2}u}{dx^{2}} = 6x$

So:

$\frac{dy}{dx} = 6u^{5}.3x^{2}\\\\\frac{dy}{dx} = 18u^{5}x^{2}\\\\$

And:

$\frac{d^{2}y}{dx^{2}} = 30u^{4}.6x\\\\\frac{d^{2}y}{dx^{2}} = 180u^{4}x$

Finally, substitute u for its equivalent expression in x:

$\frac{d^{2}y}{dx^{2}} = 180x.(x^{3} +8)^{4}$

Answer from: Quest

Yes the slope to this is -5

Answer from: Quest

mariah is studying the life cycle of the monarch butterfly. she must choose one stage in the cycle to present in science class: egg, caterpillar, chrysalis, or adult. she plans to make her choice randomly using a custom spinner divided into four sections. first, however, she spins the spinner 50 times to see the frequencies it generates.

stage times landed on

egg 10

caterpillar 12

chrysalis 15

adult 13

step-by-step explanation:

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