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Using chain rule find $ \large\frac{dw}{dt}$ for each of the following$\;w=xy+z$ where $x=\cos t,y=\sin t $

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  • Chain rule: If $u= f(x,y)$ is differentiable and $x,y $ are functions of t then $\large\frac{du}{dt}=\frac{\partial u}{\partial x}\frac{dx}{dt}+\frac{\partial u}{\partial y} \frac{dy}{dt}$
  • The chain rule for $u(x,y,z)$ where $x,y,z$ are functions of t is similarly stated.
$w=xy+z\;x=\cos t,\;y=\sin t\;z=t$
Step 1:
$\large\frac{dx}{dt}$$=-\sin t,\large\frac{dy}{dt}$$=\cos t,$$\large\frac{dz}{dt}=$$1$
$\large\frac{\partial w}{ \partial x}=$$y,\large\frac{\partial w}{\partial y}$$=x,\large\frac{\partial w}{\partial z}$$=1$
Step 2:
$\large\frac{dw}{dt}=\frac{\partial w}{\partial x}\frac{dx}{dt}+\frac{\partial w}{\partial y} \frac{dy}{dt}$
$\qquad=-y \sin t+x \cos t +1$
$\qquad=-\sin ^2 t+\cos ^2 t+1$
$\qquad=\cos ^2 t+1=2 \cos^2t$
answered Aug 13, 2013 by meena.p
 

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