Suppose $M,N,P$ are smooth manifolds. Let $f:M\to N$ and $g:M\to P$ be smooth maps. Defined a new map $h:M\to N\times P$ such that $h(x)=(f(x),g(x))$. Is it true that $dh_x=(df_x,dg_x)$? Where $dh_x$ is the differential of of $h$ at the point $x\in M$.
I was trying using an expression to prove, that is $dh(D)(q)=D(q\circ h)$ where $dh$ is the differential map of $h$, $D$ is a tangent vector in $M$ and $q$ is any smooth real valued function from $N\times P$. But did not able to prove. Is there any counter example?