A curve has parametric equations x = 1- cos(t), y = sin(t)sin(2t). Find dy/dx.

Here we have x(t) and y(t) which are both functions of t, but we want dy/dx, which doesn't involve t, we therefore need to use the chain rule. The chain rule tells us that: dy/dx = (dy/dt) x (dt/dx).y = sin(t)sin(2t) so we must use product rule to find dy/dt:dy/dt = d/dt(sin(t))sin(2t) + d/dt(sin(2t))sin(t) = cos(t)sin(2t) + 2cos(2t)sin(t).x = 1 - cos(t), therefore dx/dt = (-1) x (d/dt(cos(t)) = (-1) x (-sin(t)) = sin(t).We want dt/dx and we know that 1 / (dx/dt) = dt/dx, therefore dt/dx = 1 / sin(t).We now know that: dy/dx = (dy/dt) x (dt/dx) = (cos(t)sin(2t) +2cos(2t)sin(2t)) / (sin(t)).We need to simplify this so that we can divide by sin(t), so to help us we will rewrite sin(2t) as 2sin(t)cos(t):dy/dx = ((cos(t)(2sin(t)cos(t)) + 2cos(2t)sin(t)) / (sin(t)) = (2sin(t)cos^2(t) + 2cos(2t)sin(t)) / (sin(t)) = 2cos^2(t) + 2cos(2t), as required.