To determine

To show: fn(x)=(x+n)ex, if f(x)=xex using mathematical induction

Answer

f(n)(x)=(x+n)ex

Explanation

Given f(x)=xex

Differentiating w.r.t x

f'(x)=xex+ex.1=ex(x+1) (using Product Rule)

f'(x)=(x+1)ex

The statement is true for n=1

Let it be true for n=K, then f(K)(x)=(x+K)ex

To show it is true for n=K+1

Differentiating w.r.t x, f(K)(x) term, we get

f(K+1)(x)=(x+K).ddx(ex)+ex.ddx(x+K)

=(x+K)ex+ex.1

=ex(x+K+1)

Conclusion:   Hence by mathematical induction, we have f(n)(x)=(x+n)ex