Question

If z=sin(x/y) , x=3t , y=5−t^2 dz/dt using the chain rule. Assume the variables are restricted to domains on which the functions are defined.

dz/dt=

Answer 1

given that 7 = sin(ay) k=3t Y= 5-ť de = lesz + on the —4) Waime, and .: 7= sin(nry I Jitterentiate both op del With respect tos' oth sin carga [ (shX) = cosu = co sery) (my) )=nx?' ople =(w)) =comin) = cos(ory) 7= sin(1/3) Differentiate both side with respect to 'y' de te= * sin(wa) = cos(ag) og (78) = cos tury) (lv) = - cos (wy

=3t Jitterentiate both side with respect to t' weget = (3+) = 3 y=5- ť Ditterentiate both sides with respect to 't' Weget Yr (5-t) = - 2t From (4) The get. det er et meget = 4 castany) (3) + (-34 ) 03 (43) cat) Tsubstituting the values = costung) + 005(203) (at* - t = 2 costing) to cus (13] (lat)