First-Order Partial Derivatives

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July 26th 2008, 04:25 PM
Paperwings

First-Order Partial Derivatives

Calculate the first-order partial derivative of the following:

http://dammserv.loopback.nu/latex/im...727_HJhCAW.png

http://dammserv.loopback.nu/latex/im...727_WMExIH.png for all (x,y) in $R^2$

I used this as a composition function and used the chain rule.
http://dammserv.loopback.nu/latex/im...727_pVjxzr.png

However, I am unsure of how to apply this formula.

First-order partial derivative of x would be
http://dammserv.loopback.nu/latex/im...727_CS1jty.png

First-order pairtal derivative of y would be
http://dammserv.loopback.nu/latex/im...727_fX75s3.png

My question is http://dammserv.loopback.nu/latex/im...727_KIdP5w.png: how to compute this? Do I do another composition+chain rule?

Thank you.
July 26th 2008, 05:19 PM
Mathstud28

Quote:

Originally Posted by Paperwings

Calculate the first-order partial derivative of the following:

http://dammserv.loopback.nu/latex/im...727_HJhCAW.png

http://dammserv.loopback.nu/latex/im...727_WMExIH.png for all (x,y) in $R^2$

I used this as a composition function and used the chain rule.
http://dammserv.loopback.nu/latex/im...727_pVjxzr.png

However, I am unsure of how to apply this formula.

First-order partial derivative of x would be
http://dammserv.loopback.nu/latex/im...727_CS1jty.png

First-order pairtal derivative of y would be
http://dammserv.loopback.nu/latex/im...727_fX75s3.png

My question is http://dammserv.loopback.nu/latex/im...727_KIdP5w.png: how to compute this? Do I do another composition+chain rule?

Thank you.

You cannot, it doesn't tell you what the function is. You have gone as far as you can

http://www.texify.com/img/%5CLARGE%5...28y%5E2%29.gif
This is analgous to saying

http://www.texify.com/img/%5CLARGE%5...%28x%29%7D.gif
.

Since in terms of the derivative in respect to x we don't care whether or not there are y's in there, this is just a normal chain rule of a function of x with some constants.
July 27th 2008, 02:21 PM
Paperwings

One more question: suppose that g has second derivative, how would I calculate the second partial derivatives of the function?
July 28th 2008, 01:47 AM
Moo

1 Attachment(s)

Hello,

Quote:

Originally Posted by Paperwings

One more question: suppose that g has second derivative, how would I calculate the second partial derivatives of the function?

It depends on what second partial derivative you want.

Attachment 7338

You'll get the first one by taking the partial derivative of df/dx with respect to y. You'll get the second one by taking the partial derivative of df/dx with respect to x.
July 28th 2008, 07:07 AM
Paperwings

My book doesn't specify, but I'm pretty sure I'm supposed to find http://dammserv.loopback.nu/latex/im...728_WTkQjM.png and http://dammserv.loopback.nu/latex/im...728_dOE5l8.png. Thank you, Moo.
July 28th 2008, 01:14 PM
Mathstud28

Quote:

Originally Posted by Paperwings

My book doesn't specify, but I'm pretty sure I'm supposed to find http://dammserv.loopback.nu/latex/im...728_WTkQjM.png and http://dammserv.loopback.nu/latex/im...728_dOE5l8.png. Thank you, Moo.

Well then do what you did again

for $f_{xx}$ just hold y constant again and use either the chain rule and product rule or a combination of both if waranted. Same thing with [tex]f_{yy}[/math ] except now x is the variable held as a constant. if it is indeed $f_{xy}$ note that if hte curve is continuous that $f_{xy}=f_{yx}$ so you would only need to compute one.