Question

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 are defined.

Answer 1

$z=(x+y)e^x$
$\begin{cases}x=3t\\y=2-t^2\end{cases}$

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