Find the first and second derivatives of the following and simplify whenever possible:
x = 9t ^ 2 - 1 : y = 3t+1
Let us find the first and second derivative of the following:
x=9t2−1, y=3t+1.x = 9t ^ 2 - 1,\ \ y = 3t+1.x=9t2−1, y=3t+1.
It follows that yx′=yt′xt′=318t=16t.y_x'=\frac{y_t'}{x_t'}=\frac{3}{18t}=\frac{1}{6t}.yx′=xt′yt′=18t3=6t1.
Then yx2′′=(16t)t′xt′=−16t218t=−1108t3.y_{x^2}''=\frac{(\frac{1}{6t})_t'}{x_t'}=\frac{-\frac{1}{6t^2}}{18t}=-\frac{1}{108t^3}.yx2′′=xt′(6t1)t′=18t−6t21=−108t31.
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