ePartialDiffCompFct

Example Find the partial derivative $z_{t}$ ．

$\displaystyle z = x^{2} + xy + 2y^{2}, x = \cos{t}, y = \sin{t}$

Answer $z$ is a function of $x$ and $y$ ． $x$ and $y$ are function of $t$ ， $z$ is affected by the change in $x$ and $y$ ， $x$ and $y$ are affected by the change in $t$ ． Then，

$\displaystyle \frac{dz}{dt} = \frac{\partial z}{\partial x}\frac{\partial x}{\partial t} + \frac{\partial z}{\partial y}\frac{\partial y}{\partial t}$

Thus,

$\displaystyle \frac{dz}{dt}$	$\displaystyle =$	$\displaystyle (2x+y)(-\sin{t}) + (x+4y)(\cos{t})$
	$\displaystyle =$	$\displaystyle (2\cos{t}+\sin{t})(-\sin{t}) + (\cos{t} + 4\sin{t})(\cos{t})$
	$\displaystyle =$	$\displaystyle -2\cos{t}\sin{t}-\sin^{2}{t}+\cos^{2}{t}+4\sin{t}\cos{t}$
	$\displaystyle =$	$\displaystyle -\sin^{2}{t}+2\sin{t}\cos{t}+\cos^{2}{t}$

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