if f(x)=(x^2)(e^x) prove by induction that the nth derivative is (x^2 +2nx +n(n-1)(e^x)

Question

thanks

Answer 1

step 1.
test if it is true for n=1
1st deriv
= x^2e^x + 2xe^x
= e^x(x^2 + 2x)

by your formula is should be
e^x(x^2 + 2(1)x + 1(0))
= e^x(x^2 + 2x) check!

(I also tested it for n=2 and 3, and it checked

step 2. assume true for n=k
i.e.
kth deriv
= e^x(x^2 + 2kx + k(k-1))

step 3.
then (k+1)th deriv should be
e^x(x^2 + 2(k+1)x + (k+1)(k))

so let's differentiate the kth deriv

(k+1)th deriv
= e^x(2x + 2k) + (x^2 + 2kx + k(k-1))e^x
= e^x[x^2 + 2kx + 2x + k(k-1) + 2k]
= e^x[x^2 + 2x(k+1) + k^2 + k]
= e^x[x^2 + 2(k+1)x + k(k+1)]

as required

Q.E.D.