Question

Find the first and second derivatives of the following and simplify whenever possible:

x = 9t ^ 2 - 1 : y = 3t+1

Answer 1

Let us find the first and second derivative of the following:

$x = 9t ^ 2 - 1,\ \ y = 3t+1.$

It follows that $y_x'=\frac{y_t'}{x_t'}=\frac{3}{18t}=\frac{1}{6t}.$

Then $y_{x^2}''=\frac{(\frac{1}{6t})_t'}{x_t'}=\frac{-\frac{1}{6t^2}}{18t}=-\frac{1}{108t^3}.$