Question

How do you differentiate the following parametric equation: `x(t)=tsqrt(t^2-1), y(t)= t^2-e^(t)`?

Answer 1

`((2t - e^t)(t^2 - 1)^(1/2))/(2t^2 - 1)`

Explanation:

For the parametric function the derivative is given by

`dy/dx = (dy/dt)/(dx/dt)`

rewriting x(t) as `x(t) = t(t^2 - 1)^(1/2) " for ease of differentiating "`

Now have a product of 2 functions, which can be differentiated using the `color(blue)" product rule "`

If f(x) = g(x).h(x) then f'(x) = g(x).h'(x) + h(x).g'(x)
`"------------------------------------------------------------"`
so x'(t) = t . `d/dt(t^2-1)^(1/2) + (t^2-1)^(1/2).d/dt(t)`

`= t. 1/2(t^2-1)^(-1/2).d/dt(t^2-1) + (t^2-1)^(1/2) .1`

`= t. 1/2(t^2-1)^(-1/2). 2t + (t^2-1)^(1/2)`

`= t^2/(t^2-1)^(1/2) + (t^2 -1)^(1/2)`

rewriting as a single fraction.

`(t^2 + t^2 -1)/(t^2 -1)^(1/2) = (2t^2 -1)/(t^2 -1)^(1/2)`
`"-----------------------------------------------------------"`

and `y'(t) = 2t - e^t`
`"------------------------------------------------"`

`rArr dy/dx = (y'(t))/(x'(t)) = (2t-e^t)/((2t^2 -1)/(t^2 -1)^(1/2))`

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