Proof that a set of functions are independent

Proof that a set of functions are independent

The problem is an outline of a proof by induction for the following. Give
$r_1,\dots,r_n$ distinct real numbers, $Q_1,\dots,Q_n$ be $n$ polynomials,
none of which is the zero polynomial, prove that
$Q_1e^{r_1x},\dots,Q_ne^{r_nx}$ are independent. So the problem asks that
you do this by, knowing that it is true for $n=1$,$n=2$, assuming the
statement true for $n=p$, and then proving $\sum_{k=1}^{p+1}
c_kQ_k(x)e^{r_kx} = 0$ by multiplying the sum by $e^{-r_{p+1}x}$ and then
differentiating, then using the inductive hypothesis to show all $c_k=0$.
When I differentiate the sum I get: $$
\sum_{k=1}^{p+1}\frac{d}{dx}c_kQ_k(x)e^{(r_k-r_{p+1})x} =
\left(\sum_{k=1}^p
c_ke^{(r_k-r_{p+1})x}((r_k-r_{p+1})Q_k(x)+Q'_k(x))\right)+c_{p+1}Q'_{p+1}(x)
= 0 $$ and I'm not sure how to bring the inductive hypothesis into play.