Can Lagrange's Interpolation Be Used to Prove e^tD(f(x))=f(x+t)?

[cummings12332]

Homework Statement

Let D:R[x]->R[x]be the differentiation operator D(f(x))=f'(x),prove that
e^tD(f(x))=f(x+t) for a real number t

Homework Equations

application of Lagranges interpolation

The Attempt at a Solution

i don't know how to begin or construct the proof here

 

[Dick]

Homework Statement

Let D:R[x]->R[x]be the differentiation operator D(f(x))=f'(x),prove that
e^tD(f(x))=f(x+t) for a real number t

Homework Equations

application of Lagranges interpolation

The Attempt at a Solution

i don't know how to begin or construct the proof here

You didn't write that very grammatically. You mean e^(tD)(f(x))=f(x+t). Write out a Taylor series expansion of f(x+t) around x. Now compare it with e^(tD)=1+tD+(t^2)D^2/2!+(t^3)D^3/3!+... acting on f(x).

 

[lurflurf]

There are many ways, through most are not rigorous.

One way is to expand both side in Taylor series in t.