I'm reading about the derivative of multilinear maps
The author said that the derivative of a $m$-linear map is the sum of $(m−1)$-linear functions. But it seems to me that the each summand in the formula below is also a $m$-linear map $$\partial \varphi\left(x_{1}, \ldots, x_{m}\right)\left(h_{1}, \ldots, h_{m}\right)=\sum_{j=1}^{m} \varphi\left(x_{1}, \ldots, x_{j-1}, h_{j}, x_{j+1}, \ldots, x_{m}\right)$$
Because $\varphi\left(x_{1}, \ldots, x_{m}\right)$ is a $m$-linear map, so is $\varphi\left(x_{1}, \ldots, x_{j-1}, h_{j}, x_{j+1}, \ldots, x_{m}\right)$.
Could you please explain what I'm missing? Thank you so much!