Question

If z=e^x*sin(y), where x=st^2 and y=s^2t, find dz/ds and dz/dt in terms of s and t using the chain rule.

Thankyou!

Answer 1

The total differentials for x and y:
dx= t^2 ds +2ts dt; thus dx/ds=t^2 and dx/dt=2ts
dy= 2st ds +s^2 dt; thus dy/ds=2st and dy/dt=s^2

d(e^x)/ds= e^x *dx/ds = e^(st^2) *t^2
d(e^x)/dt= e^x *dx/dt = e^(st^2) *2ts

d(sin(y))/ds= cos(y)* dy/ds = cos(ts^2) *2st
d(sin(y))/dt= cos(y)* dy/dt = cos(ts^2) *s^2

dz/ds = d(e^x)/ds+d(sin(y))/ds = e^(st^2)*t^2+cos(ts^2)*2st
dz/dt = d(e^x)/dt+d(sin(y))/dt = e^(st^2)*2ts+cos(ts^2)*s^2