Assume x and y are functions of t. Evaluate dy/dt for 3xe^y=9-ln729+6lnx, with the conditions dx/dt=9, x=3, y=0.
To find dy/dt, we first need to differentiate both sides of the equation with respect to t using the product rule and chain rule for differentiation:
Given: 3xe^y = 9 - ln(729) + 6ln(x)
Differentiating both sides with respect to t:
d/dt(3x e^y) = d/dt(9 - ln(729) + 6ln(x))
Using the product rule and chain rule:
3(e^y(dx/dt) + x(e^y)(dy/dt) = 0 + 0 + 6(1/x)(dx/dt)
Substitute the given values:
3(e^0(9) + 3(e^0)(dy/dt) = 6(1/3)(9)
3(1)(9) + 3(dy/dt) = 6(3)
27 + 3(dy/dt) = 18
3(dy/dt) = 18 - 27
3(dy/dt) = -9
dy/dt = -3
Therefore, dy/dt = -3.