Implicit Differentition

Question

If 2x² + y² = 17 then evaluate  when x = 2 and y = 3. Round your answer to 2 decimal places. Use the hyphen symbol, -, for negative values.

Answer 1

2x² + y² = 17

4x + 2y * y' = 0

2y * y' = -4x

y' = -2x / y = -2x * y^-1

y" = -2y^-1 - 2x(-y^-2 )y'

replace y' with -2x*y^-1

y" = -2y^-1 - 2x(-y^-2 )(-2x*y^-1

y" = -2/y - 4x² / y³

evaluate at (2,3):

y" = -2/3 - 16/27 = -34/27 = -1.26

CHECK this using a graphing calculator or online graphing tool

Answer 2

2x² + y² = 17

Start by taking d/dx of both sides:

4x + 2y·y' = 0

2y·y' = -4x

y' = -4x/(2y)

y' = -2x/y

Now find y''

y'' = [y·d/dx(-2x)-(-2x)·d/dx(y)]/y²

y'' = (-2y + 2xy')/y²

^{Now evaluate y'(2,3)}

^{y'(2,3) = -2(2)/3 = -4/3}

^{Now evaluate y''(2,3)}

y''(2,3) = [-2(3)+2(2)·(-4/3)]/3²

y''(2,3) = (-6-16/3)/9 = (-18/3 - 16/3)/9 = (-34/3)/9 = -34/27 ≈ -1.26