Solutions - 0260-02

Derivative functions:

$$(x^{\prime },y^{\prime })\gg \gg (2t^{-3},1-t^{-2})$$

Slope:

$$\begin{array}{c}\frac{dy}{dx}=\frac{y^{\prime }(t)}{x^{\prime }(t)}\\ \\ \gg \gg \frac{1-t^{-2}}{2t^{-3}}\gg \gg \frac{1}{2}(t^3-t)\end{array}$$

Second derivative:

$$\begin{array}{c}\frac{d^{2}y}{d{x}^2}=\frac{d}{dt}\left(\frac{dy}{dx}\right)\cdot \frac{1}{x^{\prime }(t)}\\ \\ \gg \gg \frac{1}{2}(3t^2-1)\cdot \frac{1}{2t^{-3}}\gg \gg \frac{1}{4}(3t^5-t^3)\end{array}$$

At $t=1$:

$${\frac{d^{2}y}{d{x}^2}|}_{t=1}\gg \gg \frac{1}{4}(3-1)\gg \gg \frac{1}{2}$$