Mathematics: If -y-2x^3=Y^2 then find D^2y/dx^2 at the point (-1,-2) in simplest form ^^ the main problem i am having is when I have to use the quotient rule

The answer on Ed is A!Step-by-step explanation:...

If -y-2x^3=Y^2 then find D^2y/dx^2 at the point - id2322219947100, nssjdjsididiodododod

Ответ:

willasiacochran

13.02.2022

Mathematics

$\frac{d^2y}{dx^2} = \frac{-4}{3}$

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

BracketsParenthesisExponentsMultiplicationDivisionAdditionSubtractionLeft to Right

Algebra I

Factoring

Calculus

Implicit Differentiation

The derivative of a constant is equal to 0

Basic Power Rule:

f(x) = cxⁿf’(x) = c·nxⁿ⁻¹

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

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

Quotient Rule: $\frac{d}{dx} [\frac{f(x)}{g(x)} ]=\frac{g(x)f'(x)-g'(x)f(x)}{g^2(x)}$

Step-by-step explanation:

Step 1: Define

-y - 2x³ = y²

Rate of change of tangent line at point (-1, -2)

Step 2: Differentiate Pt. 1

Find 1st Derivative

Implicit Differentiation [Basic Power Rule]: -y'-6x^2=2yy'

[Algebra] Isolate y' terms: -6x^2=2yy'+y'

[Algebra] Factor y': -6x^2=y'(2y+1)

[Algebra] Isolate y': $\frac{-6x^2}{(2y+1)}=y'$ [Algebra] Rewrite: $y' = \frac{-6x^2}{(2y+1)}$

Step 3: Differentiate Pt. 2

Find 2nd Derivative

Differentiate [Quotient Rule/Basic Power Rule]: $y'' = \frac{-12x(2y+1)+6x^2(2y')}{(2y+1)^2}$ [Derivative] Simplify: $y'' = \frac{-24xy-12x+12x^2y'}{(2y+1)^2}$ [Derivative] Back-Substitute y': $y'' = \frac{-24xy-12x+12x^2(\frac{-6x^2}{2y+1} )}{(2y+1)^2}$ [Derivative] Simplify: $y'' = \frac{-24xy-12x-\frac{72x^4}{2y+1} }{(2y+1)^2}$

Step 4: Find Slope at Given Point

[Algebra] Substitute in x and y: $y''(-1,-2) = \frac{-24(-1)(-2)-12(-1)-\frac{72(-1)^4}{2(-2)+1} }{(2(-2)+1)^2}$ [Pre-Algebra] Exponents: $y''(-1,-2) = \frac{-24(-1)(-2)-12(-1)-\frac{72(1)}{2(-2)+1} }{(2(-2)+1)^2}$ [Pre-Algebra] Multiply: $y''(-1,-2) = \frac{-48+12-\frac{72}{-4+1} }{(-4+1)^2}$ [Pre-Algebra] Add: $y''(-1,-2) = \frac{-36-\frac{72}{-3} }{(-3)^2}$ [Pre-Algebra] Exponents: $y''(-1,-2) = \frac{-36-\frac{72}{-3} }{9}$ [Pre-Algebra] Divide: $y''(-1,-2) = \frac{-36+24 }{9}$ [Pre-Algebra] Add: $y''(-1,-2) = \frac{-12}{9}$ [Pre-Algebra] Simplify: $y''(-1,-2) = \frac{-4}{3}$

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If -y-2x^3=Y^2 then find D^2y/dx^2 at the point (-1,-2) in simplest form ^^ the main problem i am having is when I have to use the quotient rule

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