Question

How do you differentiate the following parametric equation: `x(t)=-2te^t+4t, y(t)= -2t^2-3e^(t)`?

Answer 1

the first derivative, `dy/dx = dy/dt/dx/dt` for parametric equations
the reason this holds true could be seen by treating `dy, dx, and dt` as differentials (which they are) and upon dividing, `dt` cancels out and you are left with `dy/dx`

as a refresher, `d/dt(x^t) = tx^(t-1)` and `d/dt(e^t) = e^t * t'`

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`dy/dt = d/dt (-2t^2 - 3e^t) = -4t - 3e^t`

`dx/dt = d/dt (-2te^t + 4t) = -2(e^t + te^t) + 4`

`= -2e^t - 2te^t + 4`

`dy/dx = dy/dt/dx/dt = (-4t - 3e^t)/(-2e^t - 2te^t + 4)`