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

2 Answers
Apr 13, 2015

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!

Apr 13, 2015

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#.