Calle87
differentiate using the definition



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Calle87
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\[\frac{ 1 }{ \sqrt{x} }\]

bahrom7893
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it's lim as h>0 (f(x+h)f(x))/h

bahrom7893
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f(x+h) = 1/sqrt(x+h)
f(x) = 1/sqrt(x)

bahrom7893
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Do some algebra, cancel out the hs

Calle87
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\[\frac{ \frac{ 1 }{ \sqrt{x+h} }\frac{ 1 }{ \sqrt{x} } }{ h }\]

bahrom7893
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yea, now multiply by the conjugate i think.. both top and bottom

Calle87
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\[\frac{ \frac{ \sqrt{x}+\sqrt{xh}}{ \sqrt{x}\sqrt{x+h} } }{ h }\]

across
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Typo up there ^

Calle87
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oops meant to put the h on the bottom :}

bahrom7893
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it's supposed to be a minus

bahrom7893
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all the way on top in the middle

Calle87
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oh ya that to

bahrom7893
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and the other one on top must be a plus

across
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Inside the radical, too.

bahrom7893
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u mixed up the signs

Calle87
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oops!

Calle87
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ok so i got all that part, the conjugates are what get me

bahrom7893
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twiddla time then.. hang on a sec

Calle87
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\[\frac{ h }{ h \sqrt{x}\sqrt{x+h(\sqrt{x}+\sqrt{x+h})} }\]

Calle87
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h goes away

Calle87
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that square root is mess up in the bottom

bahrom7893
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yup

bahrom7893
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No, since h is 0, on the bottom u just have sqrt(x+0)

bahrom7893
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so it's 1/sqrt(x) which is correct

Calle87
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at what point do i make h = 0?

bahrom7893
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when u dont run into trouble if u do.

bahrom7893
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1/h < NO
1/(4+1) < YES