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Step2247 [10]
4 years ago
8

Z=(x+y)ex,x=3t,y=2−t2, find dz/dt using the chain rule. assume the variables are restricted to domains on which the functions ar

e defined.
Mathematics
1 answer:
luda_lava [24]4 years ago
7 0
z=(x+y)e^x
\begin{cases}x=3t\\y=2-t^2\end{cases}

\dfrac{\mathrm dz}{\mathrm dt}=\dfrac{\partial z}{\partial x}\dfrac{\mathrm dx}{\mathrm dt}+\dfrac{\partial z}{\partial y}\dfrac{\mathrm dy}{\mathrm dt}
\dfrac{\mathrm dz}{\mathrm dt}=(e^x+(x+y)e^x)(3)+e^x(-2t)
\dfrac{\mathrm dz}{\mathrm dt}=3(1+x+y)e^x-2te^x
\dfrac{\mathrm dz}{\mathrm dt}=3(1+3t+2-t^2)e^{3t}-2te^{3t}
\dfrac{\mathrm dz}{\mathrm dt}=3(3+t-t^2)e^{3t}
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