First, differentiate \(y = xe^{ax}\) once using the product rule:

\(\frac{dy}{dx} = axe^{ax} + e^{ax} = (ax + 1)e^{ax}\), so true when \(n = 1\).

Assume that \(\frac{d^k y}{dx^k} = \left( a^k x + ka^{k-1} \right) e^{ax}\).

Differentiate the \(k\)-th derivative:

\(\frac{d^{k+1} y}{dx^{k+1}} = a \left( a^k x + ka^{k-1} \right) e^{ax} + e^{ax} \left( a^k \right) = \left( a^{k+1} + (k+1)a^k \right) e^{ax}\).

So true when \(n = k + 1\). By induction, true for all positive integers \(n\).