Question

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

Answer 1

`dx/dt = (e^t)/(4t^2) - (e^t)/(2t^3) - 1`, `dy/dt = 1 - e^t`

Explanation:

Because the curve is expressed in terms of two functions of `t` we can find the answer by differentiating each function individually with respect to `t`. First note that the equation for `x(t)` can be simplified to:

`x(t) = 1/4 e^t 1/(t^2) - t`

While `y(t)` can be left as:

`y(t) = t - e^t`

Looking at `x(t)`, it is easy to see that the application of the product rule will yield a quick answer. While `y(t)` is simply standard differentiation of each term. We also use the fact that `d/dx e^x = e^x`.

`dx/dt = (e^t)/(4t^2) - (e^t)/(2t^3) - 1`

`dy/dt = 1 - e^t`