what is the derivative of the arcsec(2t^.5)

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- Nov 17th 2009, 04:12 PMjanelle01derivative of arcsec
what is the derivative of the arcsec(2t^.5)

- Nov 18th 2009, 05:14 AMtom@ballooncalculus
Just in case a picture helps...

http://www.ballooncalculus.org/asy/diffChain/arcsec.png

... where

http://www.ballooncalculus.org/asy/chain.png

... is the chain rule. Straight continuous lines differentiate downwards (integrate up) with respect to x, and the straight dashed line similarly but with respect to the dashed balloon expression (the inner function of the composite which is subject to the chain rule).

OK, but why does

$\displaystyle \frac{d}{dx}$ arcsec $\displaystyle x = \frac{1}{x \sqrt{x^2 - 1}}$

?

This time we can make an 'internal' substitution, replacing x with sec \theta...

http://www.ballooncalculus.org/asy/t...nal/arcsec.png

As before, it's basically a clockwise journey...

http://www.ballooncalculus.org/asy/maps/inverse.png

Hope this helps.

Edit:

Alternatively, and probably much better, combine both in one... (spoiler to accommodate width)

__Spoiler__:

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Don't integrate - balloontegrate!

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