Sorry, this might look like a simple question. I need help regarding an equation $f(c)=g(x+cy)$. Here, $f$ is just a single variable function from $R$ to $R$, and $g$ is from $R^n$ to $R$, so that $x,y$ are in $R^n$. It is known that $g$ is differentiable. How do we know that $f'(c)=y^t\nabla g(x+cy)$?
My question is also about the definition of differentiability and derivative. My multivariable calculus knowledge is weak, and I went back and forth to Wikipedia but could not find a light.
When I think about it, it is just like $df/dt=\frac{d(g(x+ty))}{d(x+ty)}\cdot \frac{d(x+ty)}{dt}=\nabla g(x+ty)\cdot y$, which is as desired at $t=c$. But, why though? I am not even sure I am writing it correctly. There are so much that I like to ask.
- What is even $\frac{d}{dt}$ of $g(x+cy)$?
- What is even $\frac{d(g(x+ty))}{d(x+ty)}$? Not even sure why it is just $\nabla g(x+ty)$.
- What is even $\frac{d(x+ty)}{dt}$?
- Why would we have the dot product? I mean, of course it is more than just for the sake of matching the operation for vectors, right? There should be a definition I am missing here.
It seems like I do not know what differentiation or derivative, and chain rule are in higher dimensions. I only know the gradient $\nabla$. I only use the ordinary undergraduate $d/dt$ on the LHS, but on the middle and the RHS are something else.
Thank you for the guidance!