here's the prob

Prove if K is a normal subgroup of a group G then $\displaystyle K'\trianglelefteq G$ where $\displaystyle K'=[K,K] =\langle [a, b]| a, b \in K\rangle$.

I have a theorem i feel like i should be able to use but i'm not sure how...this is it

Let $\displaystyle H\leq G$ and let $\displaystyle x, y\in G$ then $\displaystyle H\trianglelefteq G \iff [H,G]=\leq H$

and

if $\displaystyle G'\leq H$ then $\displaystyle H\trianglelefteq G$.

any ideas?