Question

What is the nth derivative of `f(x) = x e^(2x)`?

Answer 1

We can write the derivative of this function in a ricorsive way.
Ricorsive way means that also in the definition there is the concept of derivate!

`y^((1))=1*e^(2x)+x*e^(2x)*2=e^(2x)+2xe^(2x)=e^(2x)+2y`

`y^((2))=2e^(2x)+2y^((1))`

`y^((3))=2^2e^(2x)+2y^((2))`

...

`y^((n))=2^(n-1)e^(2x)+2y^((n-1))`.

I hope it is the answer you wanted!

Answer 2

Thank you for providing a fun question to work on!

`f(x) = xe^(2x)`
`f'(x) = e^(2x) + 2xe^(2x) = e^(2x)(2x+1)`
`f''(x) = 2e^(2x)(2x+1) +e^(2x)*2 = 2e^(2x)(2x+2)`
`f'''(x) = 4e^(2x)(2x+2) + 2e^(2x)*2 = 4e^(2x)(2x+3)`

`f^((4))(x) = 8e^(2x)(2x+3) + 8e^(2x) = 8e^(2x)(2x+4)`

Looks like

`f^((n))(x) = 2^(n-1)e^(2x)(2x+n)`

It should be straightforward to prove by induction on `n`.