Math Help - Induction

1. Induction

Prove by induction the nth derivative of e^(a*x) is (a^n)*e^(a*x) for all n that exist in the natural numbers greater than or equal to 1.

That was the conjecture I made from the taking a few derivatives.

After showing p(1) is true and assuming p(k) is true, how do I start proving p(k+1) is true.

I thought about starting with p(k) and multiplying by a but I don't think that will work in this situation since the derivative must be taking.

2. Hi

You assume that p(k) is true which means $\frac{d^k e^{ax}}{dx^k} = a^k e^{ax}$

You want to show that p(k+1) is true : $\frac{d^{k+1} e^{ax}}{dx^{k+1}} = a^{k+1} e^{ax}$

Just take the derivative of $\frac{d^k e^{ax}}{dx^k} = a^k e^{ax}$