Let z=e^(x^2+3y)+x^3y^2, where x=tcos r and y = rt^4. Use the chain rule fot multivariable functions (Cals III Chain Rule) to find (delz)/(delr) and (delz)/delt). Give your answers in terms of r and t only. Be sure to show all of your work.

Maiclubk 2021-01-05 Answered
Let z=ex2+3y+x3y2, where x=tcosrandy=rt4. Use the chain rule fot multivariable functions (Cals III Chain Rule) to find zr and zt). Give your answers in terms of r and t only. Be sure to show all of your work.
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Expert Answer

Clelioo
Answered 2021-01-06 Author has 88 answers

z=ex2+3y+x3y2 (1) where x=tcosr and y=rt4 Now use the chain rule for multivariable function then (z)/(r)=(z)/(x)×(x)/(r)+(z)/(y)×(y)/r)(z)/(r)=((ex2+3y+x3y3))/(x)×((t cosr))/(r)+((ex2+3y+x3y3))/(y)×((rt4))/(r)(z)/(r)=(ex2+3y2x+3x2y2)×(tsinr)+(e(x2+3y)2x+3x2y2)×t4(z)/(r)=tsinr(2xex2+3y2x+3x2y2)+t4(3ex2+3y2x+3x3y) Now put x=tcosr,y=rt4 then (z)/(r)=tsinr(2tcos re(tcosr)2+3rt4+3(tcosr)2(rt4)2)+t4(3e(tcosr)2+3rt4)+2(tcosr)3rt4)(z)/(r)=(2t2sinrcosrr(tcosr)2+3rt4)+3r2t11sinrcos2r)+(3t4e(tcosr)2+3rt4)+2rt11cos3r)(z)/(r)=(tcosr)2+3rt4)(t2sin2r+3t4)3r2t11sinrcos2r+2rt11cos3r[,sin2x=2sinxcosx]

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