Question #178969

z=e^(x-2y) x=cost y=sint dz/dt=?


1
Expert's answer
2021-04-13T16:28:14-0400

Given: z=ex2y,x=cost,y=sintz=e^{x-2y},x=cost,y=sint

Require to find dzdt\frac{dz}{dt}


Recollect the following Chain Rule:


If z=f(x,y),x=g(t),y=h(t)z=f(x,y),x=g(t),y=h(t) , then dzdt=zxdxdt+zydydt\frac{dz}{dt}=\frac{\partial z}{\partial x}\frac{dx}{dt}+\frac{\partial z}{\partial y}\frac{dy}{dt}


Now z=ex2yzx=ex2y(12(0))=ex2yz=e^{x-2y}\Rightarrow \frac{\partial z}{\partial x}=e^{x-2y}(1-2(0))=e^{x-2y} , zy=ex2y(02(1))=2ex2y\frac{\partial z}{\partial y}=e^{x-2y}(0-2(1))=-2e^{x-2y}


And x=cost,y=sintdxdt=sint,dydt=costx=cost,y=sint\Rightarrow \frac{dx}{dt}=-sint, \frac{dy}{dt}=cost


Using the Chain Rule, we get

dzdt=zxdxdt+zydydt\frac{dz}{dt}=\frac{\partial z}{\partial x}\frac{dx}{dt}+\frac{\partial z}{\partial y}\frac{dy}{dt}

dzdt=(ex2y)(sint)+(2ex2y)(cost)\Rightarrow \frac{dz}{dt}=(e^{x-2y})(-sint)+(-2e^{x-2y})(cost)

dzdt=sintex2y2costex2y\Rightarrow \frac{dz}{dt}=-sinte^{x-2y}-2coste^{x-2y}

dzdt=ex2y[sint+2cost]\Rightarrow \frac{dz}{dt}=-e^{x-2y}[sint+2cost]


Therefore dzdt=(sint+2cost)ex2y\Rightarrow \frac{dz}{dt}=-(sint+2cost)e^{x-2y}


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