Question #7d059

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Answer 1 · 2017-10-26T04:31:15

$\frac{d y}{d x} = \frac{sin t}{1 - cos t}$ $dy/dx=sint/(1-cost)$

We have the functions

$x = t - sin t$ $x=t-sint$ and $y = 1 - cos t$ $y=1-cost$

If we find out $\frac{d y}{d t}$ $dy/dt$ and $\frac{d x}{d t}$ $dx/dt$ we can divide them and find $\frac{d y}{d x}$ $dy/dx$

$\frac{\frac{d y}{d t}}{\frac{d x}{d t}}$ $(dy/dt)/(dx/dt)$ $\to$ $->$ $(dy/canceldt)/(dx/canceldt)$ $->$ $dy/dx$

$color(red)1.$ $y=1-cost$

Differentiate both sides with respect to $t$

$d/dty=d/dt(1-cost)$

$dy/dt=0-(-sint)$

$dy/dt=sint$ $->$ $color(red)A$

$color(red)2.$ $x=t-sint$

Differentiate both sides with respect to $t$

$d/dtx=d/dt(t-sint)$

$dx/dt=1-cost$ $->$ $color(red)B$

Now we divide $color(red)A$ and $color(red)B$

$(dy/dt)/(dx/dt)=sint/(1-cost)$

$dy/dx=sint/(1-cost)$