Prove by induction that $\frac{d^ny}{dx^n} = n3^{n-1}e^{3x}+x3^ne^{3x}$ for the equation $y(x)=xe^{3x}$

Question

I have been solving this question, and I proved n=1, assumed n=k is correct and attempted to solve n=k+1. I got to the point where $$\frac{(d^{k+1}y)}{dx^{k+1}} = \frac{d⋅d^ky}{dx⋅dx^k}$$. Although I got this, I don't know how I can combine the n=k equation and $$\frac{d}{dx}$$. Can someone help me solve this problem?

Answer 1

$$\frac{d^{k+1}y}{dx^{k+1}} = \frac{d}{dx}\left(\frac{d^ky}{dx^k}\right)$$ $$ = \frac{d}{dx}\left(k3^{k-1}e^{3x} + x3^ke^{3x}\right)$$ $$= k3^ke^{3x} + \left(3^ke^{3x}+x3^{k+1}e^{3x}\right)$$ $$= \left(k + 1 \right)3^ke^{3x} + x3^{k+1}e^{3x}$$ which is the same as the relation for $n = k+1$