Analysis: prove that ln(x) is a smooth function (i.e. infinitely differentiable)

[yossup]

Homework Statement

Prove that f(x) is a smooth function (i.e. infinitely differentiable)

Homework Equations

ln(x) = \int^{x}_{1} 1/t dt

f(x) = ln(x)

The Attempt at a Solution

I was thinking about using taylor series to prove ln(x) is smooth but I'm strictly told to NOT assume f(x) = ln(x) beforehand.

 

[Office_Shredder]

I'm confused. When you say you can't assume f(x)=ln(x), then how is f(x) defined?

 

[yossup]

I'm confused. When you say you can't assume f(x)=ln(x), then how is f(x) defined?

i guess i meant - we're supposed to derive that f(x) is smooth from f(x) = \int^{x}_{1} 1/t dt, rather than f(x)=ln(x)

 

[Office_Shredder]

Ok. Can you calculate f'(x) from that definition?

 

[yossup]

Ok. Can you calculate f'(x) from that definition?

f'(x) is just 1/t, so i just go about proving that 1/t is infinitely differentiable?

 

[Office_Shredder]

That's correct, except it's 1/x not 1/t

 

[yossup]

That's correct, except it's 1/x not 1/t

ah right. i know it's intuitive that 1/x is infinitely differentiable but what do we use to rigorously prove it? taylor series?

 

[Office_Shredder]

ah right. i know it's intuitive that 1/x is infinitely differentiable but what do we use to rigorously prove it? taylor series?

Well, can you calculate the derivative of 1/x? How about the second derivative of 1/x? The third derivative?

Once you see a pattern, think about a proof by induction

 

[yossup]

Well, can you calculate the derivative of 1/x? How about the second derivative of 1/x? The third derivative?

Once you see a pattern, think about a proof by induction

thank you.

 

[Deveno]

one caveat, which i feel i ought to mention:

the domain of definition *matters*.

you can define f on (0,∞), but not on any interval containing 0 (because f is undefined).

a similar restriction holds for the derivatives.

so it's not enough just to say f is smooth...you have to say WHERE f is smooth.