Show that vector field is $C^\infty$ and calculate its Jacobian matrix at $(0,1,1)$ From an introductory course on calculus on manifolds:

Show that $f : U \subset \Bbb R^3 \to \Bbb R^2$ given by $$f(x,y,z) = \begin{bmatrix} \sin(x+z) + \log(y z^2) \\ e^{x+z} + y z \end{bmatrix}$$ is $C^\infty$ and calculate its Jacobian matrix at $(0,1,1)$ where $$U = \left\{ (x, y, z) \in \Bbb R^3 \mid y, z > 0 \right\}$$


I already showed that $f$ is continuous and that it is $C^1$, but I don't really get how to use induction from here. Any help?

Hint: Only noticing that all functions involved are well defined and $C^\infty$, should be enough. For the jacobian you need to find the partial derivatives and form the following matrix evaluated at the given point $$J(0,1,1)=\left[\begin{array}{ccc} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} & \frac{\partial u}{\partial z}\\ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} & \frac{\partial v}{\partial z} \end{array} \right]_{(0,1,1)} $$ where $u=\sin(x+z)+\log(yz^2)$ and $v=e^{x+z} +yz$.
