Why is $$\dfrac{\partial}{\partial x_i}f(tx_1, \dots, tx_n)=\dfrac{\partial f}{\partial(tx_i)}(tx_1,\dots,tx_n)\cdot\dfrac{\partial(tx_i)}{\partial x_i}=\dfrac{\partial f}{\partial x_i}(tx_1,\dots,tx_n)\cdot\dfrac{\partial(tx_i)}{\partial x_i}$$ here ? I think it is NOT equal as, I have this answer, and according to it, I should go this way: $$\begin{align}\dfrac{\partial f}{\partial(tx_i)}(tx_1,\dots,tx_n)\cdot\dfrac{\partial(tx_i)}{\partial x_i}&=\dfrac{\partial f}{\partial(tx_i)}(tx_1,\dots,tx_n)\cdot\dfrac{\partial x_i}{\partial x_i}\cdot\dfrac{\partial(tx_i)}{\partial x_i}\\&=\dfrac{\partial f}{\partial x_i}(tx_1,\dots,tx_n)\cdot\dfrac{\partial x_i}{\partial(tx_i)}\cdot\dfrac{\partial(tx_i)}{\partial x_i}\\&=\dfrac{\partial f}{\partial x_i}(tx_1,\dots,tx_n)\cdot\dfrac{t}{t}\\&=\dfrac{\partial f}{\partial x_i}(tx_1,\dots,tx_n)\\& \neq\dfrac{\partial f}{\partial x_i}(tx_1,\dots,tx_n)\cdot\dfrac{\partial(tx_i)}{\partial x_i}\end{align}$$
Please help! I am stuck in my Microeconomics course.
This has nothing to do with partial derivatives, but instead it is confusion about the chain rule.
Consider $f(g(x))$. We have $$\frac{d}{dx}[f(g(x))] = \frac{df}{dx}(g(x))\frac{dg}{dx}(x) = \frac{d}{dg}[f(g(x))]\frac{dg}{dx}(x).$$
The key part is that $$\frac{df}{dx}(g(x)) = \frac{d}{dg}[f(g(x))].$$
In your situation $$\frac{df}{dx}(tx) = \frac{d}{d(tx)}[f(tx)].$$
Sorry if I have led you astray with some of my previous comments.