Use the chain rule to find ∂z∂s and ∂z∂t, where z=x2+xy+y2,x=10s+7t,y=10s+3t

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gooberthebear8326 gooberthebear8326

19-10-2017
Mathematics

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Use the chain rule to find ∂z∂s and ∂z∂t, where z=x2+xy+y2,x=10s+7t,y=10s+3t

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LammettHash LammettHash · Answer 1 · 2017-10-20T22:50:18+02:00

[tex]\dfrac{\partial z}{\partial s}=\dfrac{\partial z}{\partial x}\dfrac{\partial x}{\partial s}+\dfrac{\partial z}{\partial y}\dfrac{\partial y}{\partial s}[/tex]
[tex]\dfrac{\partial z}{\partial s}=(2x+y)(10)+(x+2y)(10)=30x+30y[/tex]
[tex]\dfrac{\partial z}{\partial s}=30(10s+7t)+30(10s+3t)=600s+300t[/tex]

[tex]\dfrac{\partial z}{\partial t}=\dfrac{\partial z}{\partial x}\dfrac{\partial x}{\partial t}+\dfrac{\partial z}{\partial y}\dfrac{\partial y}{\partial t}[/tex]
[tex]\dfrac{\partial z}{\partial t}=(2x+y)(7)+(x+2y)(3)=17x+13y[/tex]
[tex]\dfrac{\partial z}{\partial t}=17(10s+7t)+13(10s+3t)=300s+158t[/tex]