Office_Shredder said:If f(x) and g(x) are differentiable n times, then f(x)g(x) is differentiable n times (proof by induction). So if f(x) and g(x) are differentiable infinitely many times, so is f(x)g(x).
Tedjn said:x^5 is infinitely differentiable. Why? Also, if I ask you for the nth derivative of some function h(x), for any value of n, and you can give it to me, would you agree that h(x) is infinitely differentiable?
j-lee00 said:Say I have the function
[tex] f(x) = x^5ln(x) [/tex]
I can differentiate [tex]f(x)[/tex] say 6 times and I'm left with just [tex]ln(x)[/tex]. But this is not a proof!
How do I go about proving it?
Could I use the Taylor expansion?
Wizlem said:The taylor series is unique for a given function so if you find a series which fits the taylor series form, then it is the taylor series of that function. The problem with x5*ln(x) is that ln(x) doesn't have a taylor series expansion about 0.
Well, there's not really anything to prove: the Taylor series is defined as some special power series, namely whose coefficients a_n are explicitly prescribed: some constant times the n-th derivative. Otherwisre see here.icystrike said:Simply out of curousity.. Where can i find the proof for the uniqueness of taylor series of function?
A polynomial is infinitely differentiable - every derivative after the n+1st is zero, but that is immaterial.Jilvin said:Here's what I would do:
1. Every polynomial of degree n can be differentiated (n+1) times before vanishing (it is n+1 times differentiable).
There is no such thing as an infinite order polynomial.2. Thus, an infinite order polynomial is infinitely differentiable.
This is correct heuristically, and may be all the OP needs. But I would expect that the statements made in (1) and (2) would still run afoul of his/her professor's grading. The correct terms must be used. Even so, it is possible to show that the function is infinitely differentiable without resorting to power series.3. The power series expansion of ln x is of infinite degree. This expansion absorbs the x^5 term, merely creating another infinite degree expansion with each term 5 degrees higher. This combined expansion is infinitely differentiable.
The most common way to prove that f(x) is infinitely differentiable is to use the definition of a differentiable function and show that the limit of the difference quotient exists for all values of x. This can be done by taking the derivative of f(x) and showing that it exists, and then repeating the process for the resulting derivative until the limit is reached.
A function is considered differentiable if its derivative exists at every point in its domain. A function is considered infinitely differentiable if it has derivatives of all orders at every point in its domain. In other words, a function is infinitely differentiable if it can be differentiated an infinite number of times.
No, a function must be infinitely differentiable at every point in its domain in order to be considered infinitely differentiable. If a function is not differentiable at any point in its domain, it cannot be infinitely differentiable.
The Taylor series is a representation of a function as an infinite sum of its derivatives at a certain point. If a function is infinitely differentiable, its Taylor series will converge to the function at every point in its domain. Therefore, if the Taylor series of a function converges to the function at every point, the function is infinitely differentiable.
Yes, it is possible for a function to be infinitely differentiable but not analytic. A function is considered analytic if it can be represented by a power series. While all analytic functions are infinitely differentiable, the reverse is not necessarily true. A function can be infinitely differentiable, but not have a power series representation, thus making it not analytic.