# Computing the Jacobian matrix of a restricted map

Suppose I have the map $R^2\rightarrow R^3, (x,y)\mapsto (\cos x, \cos y, \sin y)$ and I want to compute the Jacobian matrix of the restriction of this map to $y=ax+b$, i.e., $(x,ax+b)\mapsto (\cos x, \cos (ax+b), \sin (ax+b))$. How do I do it? How to differentiate with respect to the variable $ax+b$?

The restriction is given by the map $\mathbb R \to \mathbb R^2$ defined by $x \mapsto (x,ax+b)$.
Then the composition is given by $$x \mapsto (x,ax+b) \mapsto \left(\cos x , \cos (ax+b), \sin(ax+b)\right).$$
• So for example the result of differentiating $\cos(ax+b)$ w.r.t. $ax+b$ is $-a\sin(ax+b)$? – user437309 Jan 31 '18 at 19:17