Question

Assume x and y are functions of t Evaluate dy/dt for 3xe^y = 9 - ln 729 + 6 ln x, with the conditions dx/dt = 6, x = 3, y = 0 dy/dt = (Type an exact answer in simplified form.)

Answer 1

To evaluate dy/dt for the given equation, we differentiate both sides of the equation and solve for dy/dt using the given values of dx/dt, x, and y.

Step-by-step explanation:

To evaluate dy/dt for the given equation, we need to differentiate both sides of the equation with respect to t. Let's start with differentiating the left side of the equation:

Using the chain rule, we have:

(3x)(d/dt(e^y)) + e^y(d/dt(3x)) = 0

Simplifying further, we have:

3xe^y(dy/dt) + 3e^y(dx/dt) = 0

Substituting the given values of dx/dt = 6, x = 3, and y = 0, we can solve for dy/dt:

3(3)(e^0)(dy/dt) + 3(e^0)(6) = 0

Simplifying, we get:

dy/dt = -6/9 = -2/3

Answer 2

To evaluate dy/dt, we differentiate the given equation with respect to t using the chain rule and implicit differentiation. Plugging in the given values, we find that dy/dt = -6/9.

Step-by-step explanation:

To evaluate dy/dt, we need to differentiate the given equation with respect to t. We'll use the chain rule and implicit differentiation to find the derivative.

Starting with 3xe^y = 9 - ln 729 + 6 ln x:

Differentiating both sides with respect to t:

3x(e^y)(dy/dt) + 3e^y(dx/dt) = 0 + 0 + 6(x^(-1))(dx/dt)

Plugging in the given values: x = 3, y = 0, dx/dt = 6

3(3)(e^0)(dy/dt) + 3(e^0)(6) = 6(3^(-1))(6)

Simplifying further, we have: 9(dy/dt) + 18 = 36/3

9(dy/dt) + 18 = 12

9(dy/dt) = 12 - 18

9(dy/dt) = -6

Finally, solving for dy/dt gives us: dy/dt = -6/9