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&gt;&gt; ydx + 1/(xy) * ydx + xdy + 1/(xy) * xdy<br />
<br />
ok.<br />
<br />
Now, remember that x=e^t, and y=e^(-t).<br />
<br />
What's dx/dt and dy/dt then ?
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>> ydx + 1/(xy) * ydx + xdy + 1/(xy) * xdy

ok.

Now, remember that x=e^t, and y=e^(-t).

What's dx/dt and dy/dt then ?

Well, dx/dt = e^t, and dy/dt = -e^(-t), no?

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Here's a way to take less derivatives:<br />
<br />
x(t) * y(t) = e^t * e^(-t) = e^(t-t) = e^(0) = 1<br />
<br />
w(t) = xy + ln(xy) = 1 + ln 1 = 1 + 0 = 1<br />
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i.e., w(t) = 1<br />
so, w'(t) = 0
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Here's a way to take less derivatives:

x(t) * y(t) = e^t * e^(-t) = e^(t-t) = e^(0) = 1

w(t) = xy + ln(xy) = 1 + ln 1 = 1 + 0 = 1

i.e., w(t) = 1
so, w'(t) = 0

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Hi, I need to find dw/dt for w=xy + ln(xy), x=e^t, and y=e^(-t). &nbsp;I guess I go ahead and add the derivatives of x and y, giving --&gt; &nbsp; ydx + 1/(xy) * ydx + xdy + 1/(xy) * xdy = 1 + 1 -1 -1 = 0 (???)
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Hi, I need to find dw/dt for w=xy + ln(xy), x=e^t, and y=e^(-t).  I guess I go ahead and add the derivatives of x and y, giving -->   ydx + 1/(xy) * ydx + xdy + 1/(xy) * xdy = 1 + 1 -1 -1 = 0 (???)

dw/dt for a function

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