Question

Find dy/dx and d^2y/dx^2 without eliminating the parameter.

1. x=1-t^2 ; y=1+t

Answer 1

We have $x = 1 - t^2$ and $y=1+t$

We must find  $\large\frac{dy}{dx}$ and $\large\frac{d^2y}{dx^2}$ without eliminating the parameter t

$\large\frac{dy}{dx} = \large\frac{(1+t)'}{(1-t^2)} = \large\frac{1}{-2t}$

from this  $x = 1 - t^2$ , we find $t = \sqrt{1-x}$

$\large\frac{dy}{dx} = -\large\frac{1}{2\sqrt{1-x}}$

$\large\frac{d^2y}{dx^2} =$ $( -\large\frac{1}{2\sqrt{1-x}})' = \large\frac{1}{4\sqrt{(1-x)^3}}$