First, we need to define the symbolic variables and the equation. The equation in this case is 3 * e^y + x = y * tan(y) + x*y = 20.
Next, we'll differentiate the equation with respect to x. Differentiation is a process in calculus that finds the rate of change of one quantity in terms of another. It involves finding a derivative, which measures how a function changes as its input changes. In effect, we're finding the slope of the function at any point.
In this case, we find that the derivative of our equation, when differentiated with respect to x, is 1 - y.
Finally, we need to evaluate the derivative at y = 1. 'Evaluating' simply means finding the value of the derivative when y = 1. So, we substitute y = 1 into the derivative equation to get the final value for this specific condition.
So we find that the derivative at y = 1 is 0.
Comparing the derivative 1 - y with the given options, it becomes clear that the correct choice matches with (b) dy/dx = 1 - e^y, which suits the requirement that the derivative is 1 - y. This is because, in the options given, y is replaced with e^y.
So, the correct answer is (b) dy/dx = 1 - e^y.