Let $f:\mathbb{R}^n\rightarrow\mathbb{R}$ be differentiable and $t\in\mathbb{R}$.
Now, in a proof, that I have just read, the following equality was used: $$\frac{d}{dt}f(tx)=\sum x_i\frac{\partial f}{\partial x_i}(tx).$$
However, using the chain rule I get the following: $$\frac{d}{dt}f(tx)=\sum x_i\frac{\partial f}{\partial (tx_i)}(tx).$$ I subsituted $u(x):=tx$, wrote $f(tx)=f\circ u(x)$ and used the chain rule to obtain this. I do not understand how you would get the first equation. Can somebody please explain?
There is no such thing as $${\partial f\over \partial(tx_i)}\ ,$$ but there is the partial derivative of $f$ with respect to its first variable, evaluated at $tx$: $$f_{.1}(tx)={\partial f\over\partial x_1}(tx)\ .$$ This, together with the chain rule, leads to the correct first formula in your question.