Showing the properties of differentiating an integral


by greswd
Tags: differentiating, integral, properties, showing
greswd
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#1
Jan18-13, 02:11 PM
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P: 147
How do we show that

[tex]\frac{d}{dt}\left[\int\!y\,\mathrm{d} x\right] = y\,\frac{dx}{dt}[/tex]
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Mark44
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#2
Jan18-13, 03:33 PM
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Quote Quote by greswd View Post
How do we show that

[tex]\frac{d}{dt}\left[\int\!y\,\mathrm{d} x\right] = y\,\frac{dx}{dt}[/tex]
Is this a homework problem?
HallsofIvy
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#3
Jan18-13, 03:56 PM
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Quote Quote by greswd View Post
How do we show that

[tex]\frac{d}{dt}\left[\int\!y\,\mathrm{d} x\right] = y\,\frac{dx}{dt}[/tex]
Are we to assume, here, that y and x are functions of t? If we assume that y is a function of x only (with no "t" that is not in the "x") and x is a function of t, then we an write y(x(t)).

Of course, then [tex]F(x)= \int y dt[/tex] is the function such that dF/dx= y. Given that, we have that [itex]d/dt(\int y dx)= dF/dt= (dF/dx)(dx/dt)= y(x)(dx/dt)[/itex] by the chain rule.

greswd
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#4
Jan18-13, 04:07 PM
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P: 147

Showing the properties of differentiating an integral


Quote Quote by Mark44 View Post
Is this a homework problem?
Nope. Homework questions are usually standard, and answers are all in the textbooks.
I came up with this problem just out of curiosity.


Anyway, thanks for the solution HallsofIvy
DrewD
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#5
Jan18-13, 10:57 PM
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Quote Quote by greswd View Post
Nope. Homework questions are usually standard, and answers are all in the textbooks.
I wish my textbooks had the answers!
greswd
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#6
Jan22-13, 06:57 AM
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Quote Quote by DrewD View Post
I wish my textbooks had the answers!
not the exact answers, but they all follow the same template


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