Subspace, Linear Algebra, C^n[a,b]

[Dustinsfl]

Show that C^n[a,b] is a subspace of C[a,b] where C^n is the nth derivative.

I know the set is non empty since f(x)=x exist; however, I don't know how to start either the multiplication or addition property of subspaces to confirm that C^n is a subspace.

Thanks ahead of time for any help any of you may have.

 

[JSuarez]

What's the derivative of the sum of two functions? What's the derivative of a function multiplied by a constant?

 

[Dustinsfl]

alpha*n*f^(n-1)(x)
n*(f+g)^(n-1)(x)=n*[f^(n-1)(x)+g^(n-1)(x)]=n*f^(n-1)(x)+n*g^(n-1)(x)

That works for generalization all nth derivative functions?

 

[JSuarez]

Your expressions don't make much sense. If I'm reading them right (and please, try to use Latex), you're saying that:

$$\left(\alpha f\left(x\right)\right)^{n}=n\alpha\left(\alpha f\left(x\right)\right)^{n-1}$$

That doesn't make much sense.

 

[Dustinsfl]

multiplication: $$\alpha$$*n*$$f^{n-1}(x)$$
addition: n*$$(f+g)^{n-1}$$ = n*[$$f^{n-1}(x)$$+$$g^{n-1}(x)$$]
and then the n distributes.

Does this generalize all nth derivatives?

Is there away for me to import from Maple since I have Maple and Latex is to slow and cumbersome?

 

[JSuarez]

Those expressions are wrong, given functions f,g and a constant c, their n-th derivatives are:

$$\left(f+g\right)^{\left(n\right)}\left(x\right)=f^{\left(n\right)}\left(x\right)+g^{\left(n\right)}\left(x\right)$$

And:

$$\left(cf\right)^{\left(n\right)}\left(x\right)=cf^{\left(n\right)}\left(x\right)$$

 

[Dustinsfl]

But when the derivative is taking, don't the functions need to be multiplied by n and then the derivative is n-1?

 

[JSuarez]

You are confusing the derivative of a power with the n-th derivative.