Given: z=ex−2y,x=cost,y=sint
Require to find dtdz
Recollect the following Chain Rule:
If z=f(x,y),x=g(t),y=h(t) , then dtdz=∂x∂zdtdx+∂y∂zdtdy
Now z=ex−2y⇒∂x∂z=ex−2y(1−2(0))=ex−2y , ∂y∂z=ex−2y(0−2(1))=−2ex−2y
And x=cost,y=sint⇒dtdx=−sint,dtdy=cost
Using the Chain Rule, we get
dtdz=∂x∂zdtdx+∂y∂zdtdy
⇒dtdz=(ex−2y)(−sint)+(−2ex−2y)(cost)
⇒dtdz=−sintex−2y−2costex−2y
⇒dtdz=−ex−2y[sint+2cost]
Therefore ⇒dtdz=−(sint+2cost)ex−2y
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