Final answer:
To find dy/dt, we performed implicit differentiation on the equation xy = 12, substituted the given values x = 2 and dx/dt = 7, and solved for dy/dt to obtain the answer -21/2.
Step-by-step explanation:
The student's question involves applying the concept of implicit differentiation to find dy/dt given the equation xy = 12, when x = 2 and dx/dt = 7.
To solve this, differentiate both sides of the equation with respect to t, applying the product rule on the left side:
d/dt(xy) = d/dt(12)
x(dy/dt) + y(dx/dt) = 0
When x = 2 and dx/dt = 7, and knowing xy = 12, we can solve for y:
2y = 12 => y = 6
Now substitute x, y, and dx/dt into the differentiated equation:
2(dy/dt) + 6(7) = 0
2(dy/dt) + 42 = 0
2(dy/dt) = -42
dy/dt = -21
The correct answer is B. -21/2, which is found by simplifying the last equation.