In the book of Analysis on Manifolds by Munkres, at page 44-45, he argues that
But this means that the derivative of any scalar function cannot contain a term such as $x^2$, $y^3$... because it does not make the derivative linear, but this is clearly wrong, so what am I misinterpreting ?
When we talk about differentiable functions $g$ from $\mathbb R$ into $\mathbb R$, $g'(x)$ is a number. But when $g$ is a function from $\mathbb{R}^n$ to $\mathbb{R}^m$, $g'(p)$ is a linear map, not a number. In fact, if $g$ is a map from $\mathbb R$ into $\mathbb R$ and if $a=g'(x)$, we can see this number as a linear map: the multiplication by $a$. That's the connections between these to notions of derivative.
So, if $g(x)=x^3$, $g'(x)=3x^2$. That's not a problem: $g'(a)$ is the linear map $x\mapsto 3a^2x$.The presence of that square doesn't prevent this map from being linear.