Find the indicated rate given he other information. x^2+6xy+y^2=577; dx/dt=9 when x=8 and y=9; find dy/dt.

Implicit Differentiation in terms of t. Please Help ME!

Differentiate x² + 6xy + y² = 577 with respect to t:
2x dx/dt + 6(x dy/dt + y dx/dt) + 2y dy/dt = 0.
∴ (2y + 6x) dy/dt = -(2x + 6y) dx/dt.
∴ dy/dt = -((x + 3y)/(y + 3x)) dx/dt.

Substituting x = 8, y = 9, and dx/dt = 9:
dy/dt = (-(8 + 27)/(9 + 24)) * 9.
= -(35/33) * 9.
= -105/11.

No problem.

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