The value of d²y/dx² at (3, 8) is -25/32.
Since 4x² + y² = 100, we differentiate implicitly twice with respect to x to get d²y/dx².
So, d(4x² + y²)/dx = d100/dx
d(4x²)/dx + d(y²)/dx = 0
8x + 2ydy/dx = 0
2ydy/dx = -8x
dy/dx = -8x/2y
dy/dx = -4x/y
Differentiating again with respect to x, to have d²y/dx² the second derivative . So, we have
d(dy/dx)/dx = d(-4x/y)/dx
Using the quotient rule
d(u/v)/dx = [vdu/dx - udv/dx]/v²
With u = -4x and v = y
d²y/dx² = [yd(-4x)/dx - (-4x)dy/dx]/y²
d²y/dx² = [-4y + 4xdy/dx]/y²
Substituting dy/dx into the equation, we have
d²y/dx² = [-4y + 4xdy/dx]/y²
d²y/dx² = [-4y + 4x(-4x/y)]/y²
d²y/dx² = [-4y - 16x²/y)]/y²
d²y/dx² = [-4y² - 16x²]/y³
d²y/dx² = -4[y² + 4x²]/y³
Since we require the value of d²y/dx² at (3, 8), x = 3 and y = 8.
So, substituting these values into the equation, we have
d²y/dx² = -4[y² + 4x²]/y³
d²y/dx² = -4[8² + 4(3)²]/8³
d²y/dx² = -4[64 + 4(9)]/512
d²y/dx² = -4[64 + 36]/512
d²y/dx² = -4[100]/512
d²y/dx² = -100/128
d²y/dx² = -25/32
So, the value of d²y/dx² at (3, 8) is -25/32.
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