Find the first, second and third implicit derivatives of x^2+2y^2=16
i'm pretty sure the first derivative is -x/(2y), and the second is -(2y^2+x^2)/(4y^3)

dy/dx = -x/(2y)

d²y/dx² = (-2y² − x²) / (4y³)

Step-by-step explanation:

x² + 2y² = 16

Take implicit derivative once:

2x + 4y dy/dx = 0

4y dy/dx = -2x

dy/dx = -x/(2y)

Take derivative a second time:

d²y/dx² = [ (2y) (-1) − (-x) (2 dy/dx) ] / (2y)²

d²y/dx² = (-2y + 2x dy/dx) / (4y²)

d²y/dx² = (-2y + 2x (-x/(2y))) / (4y²)

d²y/dx² = (-2y − x²/y) / (4y²)

d²y/dx² = (-2y² − x²) / (4y³)

y''' = 3(x² + 2y²)/(4y³)

Step-by-step explanation:

x² +2y² =16

2x + 4y(y') = 0

4y(y') = -2x

2y(y') = -x

y' = -x/2y

2y(y') = -1

2[y'×y' + y×y"] = -1

2(y')² + 2y(y") = -1

2(-x/2y)² + 2y(y") = -1

2(x²/4y²) + 2y(y") = -1

2y(y") = -1 - 2(x²/4y²)

8y³(y") = -4y² - 2x²

4y³(y") = -2y² - x²

y" = [-x² - 2y²] ÷ (4y³)

y" = -(x² + 2y²)/(4y³)

4y³(y") = -2y² - x²

4y³(y"') + 12y²(y')(y") = -4y(y') - 2x

4y³(y''') + 12y³[-(x² + 2y²)/(4y³)] =

-4y(-x/2y) - 2x

4y³(y''') + 3(-x²-2y²) = 2x - 2x

4y³(y''') = 3(x² + 2y²)

y''' = 3(x² + 2y²)/(4y³)

(2 x^3 + 7 x^2 + 5 x + 2)/(x (x + 2))

step-by-step explanation:

simplify the following:

x/x^2 + 3 x + 2 - (x (x + 1))/(x + 2)

combine powers. x/x^2 = x^(1 - 2):

x^(1 - 2) + 3 x + 2 - (x (x + 1))/(x + 2)

1 - 2 = -1:

x^(-1) + 3 x + 2 - (x (x + 1))/(x + 2)

put each term in 1/x + 3 x + 2 - (x (x + 1))/(x + 2) over the common denominator x (x + 2): 1/x + 3 x + 2 - (x (x + 1))/(x + 2) = (x + 2)/(x (x + 2)) + (3 x^2 (x + 2))/(x (x + 2)) + (2 x (x + 2))/(x (x + 2)) - (x^2 (x + 1))/(x (x + 2)):

(x + 2)/(x (x + 2)) + (3 x^2 (x + 2))/(x (x + 2)) + (2 x (x + 2))/(x (x + 2)) - (x^2 (x + 1))/(x (x + 2))

(x + 2)/(x (x + 2)) + (3 x^2 (x + 2))/(x (x + 2)) + (2 x (x + 2))/(x (x + 2)) - (x^2 (x + 1))/(x (x + 2)) = ((x + 2) + 3 x^2 (x + 2) + 2 x (x + 2) - x^2 (x + 1))/(x (x + 2)):

(2 + x + 3 x^2 (x + 2) + 2 x (x + 2) - x^2 (x + 1))/(x (x + 2))

3 x^2 (x + 2) = 3 x^3 + 6 x^2:

(2 + x + 3 x^3 + 6 x^2 + 2 x (x + 2) - x^2 (x + 1))/(x (x + 2))

2 x (x + 2) = 2 x^2 + 4 x:

(2 + x + 6 x^2 + 3 x^3 + 2 x^2 + 4 x - x^2 (x + 1))/(x (x + 2))

-x^2 (x + 1) = -x^2 - x^3:

(-x^2 - x^3 + 3 x^3 + 6 x^2 + 2 x^2 + 4 x + x + 2)/(x (x + 2))

grouping like terms, 3 x^3 - x^3 + 6 x^2 + 2 x^2 - x^2 + 4 x + x + 2 = (3 x^3 - x^3) + (6 x^2 + 2 x^2 - x^2) + (x + 4 x) + 2:

((3 x^3 - x^3) + (6 x^2 + 2 x^2 - x^2) + (x + 4 x) + 2)/(x (x + 2))

3 x^3 - x^3 = 2 x^3:

(2 x^3 + (6 x^2 + 2 x^2 - x^2) + (x + 4 x) + 2)/(x (x + 2))

6 x^2 + 2 x^2 - x^2 = 7 x^2:

(2 x^3 + 7 x^2 + (x + 4 x) + 2)/(x (x + 2))

x + 4 x = 5 x:

answer:   (2 x^3 + 7 x^2 + 5 x + 2)/(x (x + 2))

the answer is a

step-by-step explanation:

39.00 x 24 = total installment price

carrying installment price - cash price

needed to save the .00/