Sure, I'd be happy to walk through the solution step-by-step.
To start off, a parametric representation of a function consists of a pair of equations, x(t) and y(t), where t is a parameter.
First, let's find the derivative of each function with respect to t.
Given the function x(t) = (1/2)cos(t), the derivative can be found using the following rule: the derivative of cos(t) is -sin(t), so we have
(dx/dt) = -sin(t)/2.
Next, the derivative of the function y(t) = sin(2t) can be found as follows. Notice that we are finding the derivative of a sine function that is in terms of 2t, rather than just t, so by the chain rule, we multiply the derivative by 2. Hence, the derivative of sin(t) is cos(t), so
(dy/dt) = 2cos(2t).
Now, for a parametric curve, dy/dx, or y' is found by taking dy/dt divided by dx/dt. Hence, we have:
(dy/dx) = (dy/dt) / (dx/dt).
Substitute the derivatives we found earlier into the formula:
(dy/dx) = [2cos(2t)] / [-sin(t)/2]
By simplification:
(dy/dx) = -4cos(2t) / sin(t).
So there you have it, that's how we find dy/dx for the given parametric curve! In this case, it ended up being -4cos(2t) / sin(t).