Answered

Get the answers you need at Westonci.ca, where our expert community is dedicated to providing you with accurate information. Get immediate and reliable solutions to your questions from a community of experienced professionals on our platform. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.

Find d²y / dx², dy / dx for the curve whose equation is x² + y²_3xy_6x + 5y + 2 = 0 at the point (2,3)​

Sagot :

LammettHash

I assume you meant to write

x² + y² - 3xy - 6x + 5y + 2 = 0

Taking y = y(x) and differentiating both sides with respect to x with the power, product, and chain rules gives

2x + 2y dy/dx - 3y - 3x dy/dx - 6 + 5 dy/dx = 0

Solve for dy/dx :

2x - 3y - 6 + (2y - 3x + 5) dy/dx = 0

(2y - 3x + 5) dy/dx = 6 - 2x + 3y

dy/dx = (6 - 2x + 3y) / (2y - 3x + 5)

Now differentiate both sides again. Using the quotient and chain rules,

d²y/dx² = ((2y - 3x + 5) (-2 + 3 dy/dx) - (6 - 2x + 3y) (2 dy/dx - 3)) / (2y - 3x + 5)²

Expand and simplify the numerator:

d²y/dx² = (5y + 8 + (3 - 5x) dy/dx) / (2y - 3x + 5)²

Substitute dy/dx and simplify d²y/dx² :

d²y/dx² = (5y + 8 + (3 - 5x) (6 - 2x + 3y) / (2y - 3x + 5)) / (2y - 3x + 5)²

d²y/dx² = ((5y + 8) (2y - 3x + 5) + (3 - 5x) (6 - 2x + 3y)) / (2y - 3x + 5)³

d²y/dx² = (10y² + 50y - 30xy + 10x²  - 60x + 58) / (2y - 3x + 5)³

Thank you for visiting our platform. We hope you found the answers you were looking for. Come back anytime you need more information. Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Thank you for visiting Westonci.ca, your go-to source for reliable answers. Come back soon for more expert insights.