4y 2 1 Y 44

\text{Observe the derivative with respect to }x

\frac{d^2y}{dx^2}=\frac{-y^{2}-2y-5-ten^{2}+4x}{y^{3}+3y^{2}+3y+1}

Take the derivative of both sides

\frac{d}{dx}\left(4x^{2}+4y^{2}-16x\right)=\frac{d}{dx}\left(-8y+44\right)

\frac{d}{dx}\left(4x^{2}+4y^{2}-16x\right)

\frac{d}{dx}\left(4x^{2}\right)+\frac{d}{dx}\left(4y^{2}\right)+\frac{d}{dx}\left(-16x\right)

Evaluate

\frac{d}{dx}\left(4x^{2}\right)

\text{Utilise differentiation rule }\frac{d}{dx}\left(cf\left(ten\right)\right)=c\times\frac{d}{dx}(f(x))

four\times \frac{d}{dx}\left(10^{2}\right)

\text{Employ }\frac{d}{dx} ten^{n}=northward x^{northward-one}\text{ to find derivative}

4\times 2x

8x+\frac{d}{dx}\left(4y^{2}\right)+\frac{d}{dx}\left(-16x\right)

Evaluate

\frac{d}{dx}\left(4y^{ii}\right)

Utilize differentiation rules

\frac{d}{dy}\left(4y^{2}\correct)\times \frac{dy}{dx}

Evaluate the derivative

8y\frac{dy}{dx}

8x+8y\frac{dy}{dx}+\frac{d}{dx}\left(-16x\correct)

Evaluate

\frac{d}{dx}\left(-16x\right)

\text{Use differentiation rule }\frac{d}{dx}\left(cf\left(x\right)\right)=c\times\frac{d}{dx}(f(x))

-sixteen\times \frac{d}{dx}\left(10\right)

\text{Use }\frac{d}{dx} 10^{n}=n x^{n-1}\text{ to find derivative}

-16\times ane

8x+8y\frac{dy}{dx}-16

8x+8y\frac{dy}{dx}-16=\frac{d}{dx}\left(-8y+44\right)

\frac{d}{dx}\left(-8y+44\right)

\frac{d}{dx}\left(-8y\right)+\frac{d}{dx}\left(44\right)

Evaluate

\frac{d}{dx}\left(-8y\right)

Use differentiation rules

\frac{d}{dy}\left(-8y\right)\times \frac{dy}{dx}

Evaluate the derivative

-8\frac{dy}{dx}

-8\frac{dy}{dx}+\frac{d}{dx}\left(44\right)

8x+8y\frac{dy}{dx}-sixteen=-8\frac{dy}{dx}

8x-16+8y\frac{dy}{dx}=-viii\frac{dy}{dx}

Move the variable to the left side

8x-16+8y\frac{dy}{dx}+viii\frac{dy}{dx}=0

8y\frac{dy}{dx}+8\frac{dy}{dx}

\left(8y+8\right)\times \frac{dy}{dx}

\left(8y+eight\correct)\frac{dy}{dx}

8x-16+\left(8y+8\correct)\frac{dy}{dx}=0

Movement the constant to the correct side

\left(8y+viii\right)\frac{dy}{dx}=0-\left(8x-sixteen\right)

\left(8y+eight\right)\frac{dy}{dx}=-8x+16

\frac{\left(8y+viii\right)\frac{dy}{dx}}{8y+eight}=\frac{-8x+sixteen}{8y+8}

\frac{\left(-x+2\right)\times viii}{\left(y+1\right)\times viii}

\frac{\left(8y+eight\right)\frac{dy}{dx}}{8y+8}=\frac{-x+2}{y+1}

\frac{dy}{dx}=\frac{-x+2}{y+1}

Take the derivative of both sides

\frac{d}{dx}\left(\frac{dy}{dx}\right)=\frac{d}{dx}\left(\frac{-ten+2}{y+1}\right)

\frac{d^2y}{dx^2}=\frac{d}{dx}\left(\frac{-x+ii}{y+1}\right)

Use differentiation rules

\frac{d^2y}{dx^two}=\frac{\frac{d}{dx}\left(-x+2\right)\times \left(y+one\correct)-\left(-x+2\correct)\times \frac{d}{dx}\left(y+1\correct)}{\left(y+1\right)^{2}}

\frac{d}{dx}\left(-ten+two\right)

\frac{d}{dx}\left(-x\correct)+\frac{d}{dx}\left(2\correct)

-1+\frac{d}{dx}\left(two\right)

\frac{d^2y}{dx^2}=\frac{-\left(y+1\right)-\left(-x+2\right)\times \frac{d}{dx}\left(y+1\correct)}{\left(y+1\right)^{2}}

\frac{d}{dx}\left(y+1\right)

\frac{d}{dx}\left(y\right)+\frac{d}{dx}\left(1\right)

\frac{dy}{dx}+\frac{d}{dx}\left(1\right)

\frac{d^2y}{dx^ii}=\frac{-\left(y+ane\right)-\left(-x+2\right)\frac{dy}{dx}}{\left(y+1\right)^{2}}

-\left(y+1\right)-\left(-10+2\right)\frac{dy}{dx}

-\left(y+one\right)+\left(ten-two\correct)\frac{dy}{dx}

-y-1+\left(10-two\right)\frac{dy}{dx}

-y-ane+x\frac{dy}{dx}-2\frac{dy}{dx}

\frac{d^2y}{dx^2}=\frac{-y-1+x\frac{dy}{dx}-2\frac{dy}{dx}}{\left(y+1\right)^{2}}

\text{Apply equation }\frac{dy}{dx}=\frac{-x+two}{y+1}\text{ to substitute}

\frac{d^2y}{dx^two}=\frac{-y-1+10\times \frac{-x+2}{y+one}-2\times \frac{-ten+2}{y+1}}{\left(y+i\correct)^{2}}

\frac{-y-1+ten\times \frac{-x+ii}{y+1}-two\times \frac{-10+2}{y+1}}{\left(y+1\correct)^{2}}

\frac{-y-1+\frac{x\left(-x+ii\right)}{y+1}-2\times \frac{-x+2}{y+1}}{\left(y+1\right)^{2}}

\frac{-y-1+\frac{x\left(-ten+two\right)}{y+1}-\frac{2\left(-x+2\correct)}{y+i}}{\left(y+ane\correct)^{2}}

Calculate

-y-1+\frac{x\left(-x+two\right)}{y+1}-\frac{two\left(-x+2\right)}{y+one}

Factor the expression

-y-1+\frac{-10^{ii}+2x}{y+1}-\frac{-2x+4}{y+1}

Reduce fractions to a mutual denominator

-\frac{y\times \left(y+1\right)}{y+1}-\frac{y+i}{y+ane}+\frac{-ten^{two}+2x}{y+1}-\frac{-2x+4}{y+1}

Reorder the terms

-\frac{y\left(y+ane\correct)}{y+1}-\frac{y+ane}{y+one}+\frac{-x^{two}+2x}{y+1}-\frac{-2x+4}{y+1}

Add the terms

\frac{-y\left(y+1\right)-\left(y+one\right)-x^{2}+2x-\left(-2x+iv\right)}{y+ane}

Summate

\frac{y^{2}+y-\left(y+1\right)-x^{2}+2x-\left(-2x+4\right)}{y+ane}

If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and alter the sign of every term within the parentheses

\frac{-y^{2}-y-\left(y+1\right)-x^{ii}+2x-\left(-2x+4\correct)}{y+1}

If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses

\frac{-y^{two}-y-y-one-10^{2}+2x-\left(-2x+iv\right)}{y+1}

If a negative sign or a subtraction symbol appears exterior parentheses, remove the parentheses and change the sign of every term inside the parentheses

\frac{-y^{2}-y-y-1-x^{2}+2x+2x-4}{y+1}

Summate

\frac{-y^{2}-2y-5-x^{two}+4x}{y+one}

\frac{\frac{-y^{ii}-2y-v-x^{2}+4x}{y+ane}}{\left(y+ane\right)^{two}}

Multiply by the reciprocal

\frac{-y^{two}-2y-5-x^{two}+4x}{y+one}\times \frac{1}{\left(y+one\correct)^{ii}}

\frac{-y^{2}-2y-five-ten^{2}+4x}{\left(y+1\correct)\left(y+one\correct)^{2}}

\frac{-y^{2}-2y-five-x^{2}+4x}{\left(y+one\correct)^{iii}}

\frac{-y^{2}-2y-five-ten^{two}+4x}{y^{3}+3y^{ii}+3y+one}

\frac{d^2y}{dx^two}=\frac{-y^{ii}-2y-5-ten^{2}+4x}{y^{three}+3y^{two}+3y+i}

\text{Detect the derivative with respect to }y

\frac{d^2x}{dy^2}=-\frac{x^{2}-4x+5+2y+y^{2}}{x^{iii}-6x^{2}+12x-8}

Take the derivative of both sides

\frac{d}{dy}\left(4x^{2}+4y^{ii}-16x\right)=\frac{d}{dy}\left(-8y+44\correct)

\frac{d}{dy}\left(4x^{ii}+4y^{two}-16x\right)

\frac{d}{dy}\left(4x^{2}\right)+\frac{d}{dy}\left(4y^{2}\right)+\frac{d}{dy}\left(-16x\right)

Evaluate

\frac{d}{dy}\left(4x^{two}\correct)

Use differentiation rules

\frac{d}{dx}\left(4x^{ii}\right)\times \frac{dx}{dy}

Evaluate the derivative

8x\frac{dx}{dy}

8x\frac{dx}{dy}+\frac{d}{dy}\left(4y^{2}\right)+\frac{d}{dy}\left(-16x\right)

Evaluate

\frac{d}{dy}\left(4y^{2}\right)

\text{Utilise differentiation dominion }\frac{d}{dx}\left(cf\left(x\right)\correct)=c\times\frac{d}{dx}(f(10))

4\times \frac{d}{dy}\left(y^{ii}\right)

\text{Use }\frac{d}{dx} ten^{n}=n ten^{north-i}\text{ to find derivative}

4\times 2y

8x\frac{dx}{dy}+8y+\frac{d}{dy}\left(-16x\correct)

Evaluate

\frac{d}{dy}\left(-16x\right)

Use differentiation rules

\frac{d}{dx}\left(-16x\right)\times \frac{dx}{dy}

Evaluate the derivative

-sixteen\frac{dx}{dy}

8x\frac{dx}{dy}+8y-16\frac{dx}{dy}

8x\frac{dx}{dy}+8y-16\frac{dx}{dy}=\frac{d}{dy}\left(-8y+44\right)

\frac{d}{dy}\left(-8y+44\right)

\frac{d}{dy}\left(-8y\right)+\frac{d}{dy}\left(44\correct)

Evaluate

\frac{d}{dy}\left(-8y\right)

\text{Use differentiation rule }\frac{d}{dx}\left(cf\left(x\right)\correct)=c\times\frac{d}{dx}(f(x))

-8\times \frac{d}{dy}\left(y\right)

\text{Apply }\frac{d}{dx} x^{n}=n x^{northward-i}\text{ to observe derivative}

-8\times 1

-8+\frac{d}{dy}\left(44\correct)

8x\frac{dx}{dy}+8y-16\frac{dx}{dy}=-8

\left(8x-16\right)\frac{dx}{dy}+8y=-8

Movement the constant to the right side

\left(8x-16\right)\frac{dx}{dy}=-8-8y

\frac{\left(8x-16\right)\frac{dx}{dy}}{8x-sixteen}=\frac{-8-8y}{8x-16}

-\frac{\left(ane+y\right)\times 8}{\left(x-2\correct)\times 8}

\frac{\left(8x-16\right)\frac{dx}{dy}}{8x-16}=-\frac{1+y}{x-two}

\frac{dx}{dy}=-\frac{one+y}{x-2}

Accept the derivative of both sides

\frac{d}{dy}\left(\frac{dx}{dy}\right)=\frac{d}{dy}\left(-\frac{1+y}{x-2}\correct)

\frac{d^2x}{dy^2}=\frac{d}{dy}\left(-\frac{one+y}{x-two}\right)

Use differentiation rules

\frac{d^2x}{dy^ii}=-\frac{\frac{d}{dy}\left(1+y\right)\times \left(ten-2\right)-\left(1+y\correct)\times \frac{d}{dy}\left(ten-2\correct)}{\left(x-2\right)^{two}}

\frac{d}{dy}\left(one+y\right)

\frac{d}{dy}\left(1\right)+\frac{d}{dy}\left(y\right)

0+\frac{d}{dy}\left(y\right)

\frac{d^2x}{dy^2}=-\frac{ten-ii-\left(1+y\right)\times \frac{d}{dy}\left(10-two\right)}{\left(ten-2\right)^{ii}}

\frac{d}{dy}\left(x-2\right)

\frac{d}{dy}\left(x\right)+\frac{d}{dy}\left(-2\right)

\frac{dx}{dy}+\frac{d}{dy}\left(-2\correct)

\frac{d^2x}{dy^2}=-\frac{x-2-\left(1+y\right)\frac{dx}{dy}}{\left(x-ii\right)^{2}}

\left(-i-y\right)\times \frac{dx}{dy}

Utilize the distributive holding

-1\times \frac{dx}{dy}-y\times \frac{dx}{dy}

-\frac{dx}{dy}-y\times \frac{dx}{dy}

-\frac{dx}{dy}-y\frac{dx}{dy}

\frac{d^2x}{dy^2}=-\frac{ten-two-\frac{dx}{dy}-y\frac{dx}{dy}}{\left(10-2\correct)^{ii}}

\text{Use equation }\frac{dx}{dy}=-\frac{1+y}{x-2}\text{ to substitute}

\frac{d^2x}{dy^2}=-\frac{x-2-\left(-\frac{i+y}{x-2}\correct)-y\left(-\frac{one+y}{x-2}\correct)}{\left(x-2\right)^{two}}

-\frac{ten-2-\left(-\frac{one+y}{x-2}\correct)-y\left(-\frac{1+y}{x-two}\right)}{\left(x-ii\correct)^{2}}

Evaluate

y\left(-\frac{1+y}{ten-two}\right)

Rewrite the expression

-y\times \frac{1+y}{x-two}

Multiply the terms

-\frac{y\left(1+y\correct)}{x-two}

-\frac{x-2-\left(-\frac{i+y}{x-2}\right)-\left(-\frac{y\left(1+y\right)}{x-2}\right)}{\left(x-2\right)^{2}}

Summate

x-2-\left(-\frac{one+y}{x-2}\right)-\left(-\frac{y\left(i+y\correct)}{10-2}\correct)

Calculate

x-two+\frac{1+y}{x-2}+\frac{y\left(ane+y\right)}{x-2}

Cistron the expression

x-2+\frac{i+y}{x-2}+\frac{y+y^{2}}{10-2}

Reduce fractions to a common denominator

\frac{x\times \left(10-2\right)}{x-two}-\frac{2\left(10-two\correct)}{ten-2}+\frac{1+y}{x-ii}+\frac{y+y^{2}}{x-ii}

Reorder the terms

\frac{x\left(x-ii\right)}{x-2}-\frac{2\left(x-2\right)}{x-two}+\frac{1+y}{10-two}+\frac{y+y^{two}}{x-2}

Add together the terms

\frac{x\left(10-2\correct)-2\left(x-2\right)+i+y+y+y^{2}}{x-2}

Summate

\frac{x^{2}-2x-ii\left(x-2\right)+1+y+y+y^{ii}}{x-2}

Calculate

\frac{x^{2}-2x-\left(2x-iv\correct)+1+y+y+y^{2}}{x-2}

If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and alter the sign of every term within the parentheses

\frac{x^{ii}-2x-2x+iv+ane+y+y+y^{two}}{x-2}

Calculate

\frac{ten^{2}-4x+5+2y+y^{two}}{x-2}

-\frac{\frac{10^{2}-4x+5+2y+y^{two}}{x-two}}{\left(10-2\correct)^{2}}

Evaluate

\frac{\frac{x^{2}-4x+five+2y+y^{2}}{x-2}}{\left(x-2\right)^{2}}

Multiply by the reciprocal

\frac{10^{2}-4x+five+2y+y^{2}}{x-2}\times \frac{1}{\left(x-2\right)^{2}}

Multiply the terms

\frac{x^{2}-4x+5+2y+y^{2}}{\left(x-2\right)\left(ten-2\correct)^{2}}

Multiply the terms

\frac{10^{two}-4x+5+2y+y^{2}}{\left(x-2\correct)^{3}}