Differentiate 1/√(1+x²) from first principles without getting stuck

[rsnd]

I was told to differentiate by definition! Function
f = 1/(sqrt(1+x^2));

which gives me the expression
Lim(h->0) {(1/(sqrt(1+(x+h)^2)) - 1/(sqrt(1+x^2)))/h}
problem is...I can’t seem to get rid of the h at the bottom...I’ve tried all the math packages I have including maple and all of them just seem to complicate things! what should be the simplified version of that function so I don’t get h at the bottom resulting in undefined numbers. or is there a different way to look at this definition problem?

Thanks heaps
K. Civilian

 

[matt grime]

Why would you want to get rid of it? you need to let it tend to zero...


This ought to help:


Edit:


sqrt(a)-sqrt(b)= (a-b)/(sqrt(a)+sqrt(b))

 

[uart]

I suggest you start like this :

( 1 + (x+h)^2 )^(-1/2)

= ( 1 + x^2 + (2x+h)h )^(-1/2)

= (1 + x^2)^(-1/2) ( 1 + (2x+h)/(1+x^2) h )^(-1/2)

Then do a binomial expansion of the "(1 + (2x+h)/(1+x^2) h)^(-1/2)" factor and it's pretty straight forward from there.

 

[HallsofIvy]

Why would you want to get rid of it? you need to let it tend to zero...

Because if it is in the denominator as it goes to 0 it will cause a lot of trouble!

This ought to help:

sqrt(a)-sqrt(b)= 1/(sqrt(a)+sqrt(b))

Provided a-b= 1?

 

[matt grime]

Noted my mistake.

But the point is that if you use that corrected identity, then you get something where the h on bottom cancels off, and you just have an h on top, let it go to zero and the correct derivative drops out.

Since it's easier to typeset, here's the idea of 1/x

1/(x+h) - 1/x = -h/(x+h)(x)

divide through by h and now let he tend to zero to see that the derivative is -1/x^2

that works here too, but it's a bugger to typeset.

 

[HallsofIvy]

Noted my mistake.

But the point is that if you use that corrected identity, then you get something where the h on bottom cancels off, and you just have an h on top, let it go to zero and the correct derivative drops out.QUOTE]

Yes, but I assumed that was what the orginal post meant by "get rid of the h at the bottom".