Spivak gives an example which has step that is giving me some problems to get it, even if it's supposed to be trivial.
Spivak says:
Let $f:\mathbb{R}^2\to\mathbb{R}$ given by $f(x,y)=\sin(xy^2)$. Since $f=\sin\circ(\pi_1\cdot(\pi_2)^2)$ then $$f'(a,b)=\sin'(ab^2)\cdot(b^2\pi_1'(a,b)+2ab(\pi_2)'(a,b))\\ =\cos(ab^2)\cdot(b^2(1,0)+2ab(0,1)) \\=(b^2\cos(ab^2),2ab\cos(ab^2)).$$
The second step when he gets $f'$, how the $\pi_1'(a,b)$ becomes $(1,0)$ and $\pi_2'(a,b)$ becomes $(0,1)$?. Considering the first one, if $\pi_1$ is supposed to be the projection then according to Theorem 2-3(b) in the book must be $\pi_1'=\pi_1$ because of the linearity of the function, but then shouldn't be $a$ instead of $(1,0)$ since $\pi_1:\mathbb{R}^2\to\mathbb{R}$?.
This is a standard problem with the notation in multivariable calculus/analysis. We're evaluating the derivative of $\pi_i$ at the point $(a,b)$. The derivative of a linear map $T$ at any point in its domain is still $T$. In particular, $(a,b)$ is not the vector on which we're evaluating.