Question

Найдите производную функции y' заданной параметрически:
$1) x=ln(1+ t^{2}), y=arctgt, t= frac{1}{2} 2) x=3t+1, y= t^{2}, t=3 3) x= sqrt{2t- t^{2}} , y=arcsin (t-1), t= frac{2}{3}$

Answer 1

$y'=dfrac{y'_t}{x'_t}$

1. x' = 2t/(1+t^2); y' = 1/(1+t^2)
y'/x' = 1/2t
y'(t=1/2) = 1
2. x'=3; y'=2t
y'(t=3) = 2*3/3=2
3. x'(t) = (2 - 2t) / 2sqrt(2t-t^2) = (1 - t) / sqrt(2t - t^2)
y'(t) = 1 / sqrt(1 - (t-1)^2) = 1 / sqrt(2t - t^2)
y'(t=2/3) = 1 / (1 - 2/3) = 3