How to show $\frac{du^k}{dt}=\frac{du}{dt}\wedge u^{k-1}$?

Question

Let $E$ be a $\mathbb R$-vector space of finite dimension and $u:\mathbb R\longrightarrow \Lambda^p E$ a differentiable map where $\Lambda^p E$ is the $p$-th exterior power of $E$. How to show that $$\frac{du^k}{dt}=\frac{du}{dt}\wedge u^{k-1}.$$

This was in a book of multilinear algebra. I suppose the notation $u^k$ means $u=u\underbrace{\wedge\ldots \wedge}_{k} u$.

Thanks.