Let $F:\mathbb{R}^3\to\mathbb{R}^3$ be of class $C^\infty$.
I am trying to solve this exercise from my calculus class: Let $s\in\mathbb{R}$ and compute $$\frac{\partial}{\partial s} \left(\sum_{j=1}^3 \frac{\partial F_1}{\partial x_j}(s y), \sum_{j=1}^3 \frac{\partial F_2}{\partial x_j}(s y), \sum_{j=1}^3 \frac{\partial F_3}{\partial x_j}(s y)\right),$$ where $y=(y_1, y_2, y_3)\in\mathbb{R}^3$.
As I am not sure how to proceed, I am starting considering the term $$\frac{\partial}{\partial s} \frac{\partial F_1}{\partial x_j}(s y).$$ I think it should be $\frac{\partial^2 F_1}{\partial x_1 x_k}$ with $k=2, 3$, i.e. the first column of the Hessian matrix of $F_1$. Anyway, even if this was correct, I don't know how to upgrade to the more challenging case considered the given exercise.
Could someone please help? How the result is related to the Hessian matrix of each component $F_i$? Thank you.
We have:
$$\frac{\partial}{\partial s} \frac{\partial F_1}{\partial x_j}(s y) = \sum_{i=1}^3 y_i\frac{\partial^2 F_1}{\partial x_i x_j}$$ Thus: $$\frac{\partial}{\partial s} \left(\sum_{j=1}^3 \frac{\partial F_1}{\partial x_j}(s y)\right) = \text{sum of components of vector }y*H_1 = y*H_1*(1,1,1)^T$$ where $H_1$ is the Hessian of $F_1$. Now proceed by computing your original derivative componentwise.