### Home > APCALC > Chapter 12 > Lesson 12.2.1 > Problem12-72

12-72.

The position of a particle moving in the $xy$-plane is given by the following parametric function: $x(t) = \frac { 5 - 6 } { 3 t - 6 } ,$ $y ( t ) = \frac { 3 t } { 2 t + 6 }$

1. Determine the slope of the line tangent to the path of the particle when $t = 3$.

$\frac{dy}{dx}\Big|_{t=3}=\frac{y^\prime(3)}{x^\prime(3)}$

2. Write and interpret $\frac { d ^ { 2 } y } { d x ^ { 2 } }$

$\frac{d^2y}{dx^2}=\frac{d}{dx}\Big(\frac{dy}{dx}\Big)=\frac{d}{dx}\Big(\frac{y^\prime(t)}{x^\prime(t)}\Big)=\frac{dt}{dx}\cdot\frac{d}{dt}\Big(\frac{y^\prime(t)}{x^\prime(t)}\Big)=\frac{\frac{d}{dt}\Big(\frac{y^\prime(t)}{x^\prime(t)}\Big)}{\frac{dx}{dt}}$