I'm a beginner in this type of math, we are just starting to study it, but I need some clarifications about the meaning and the difference of when we write
$$\partial_i \qquad \text{and}\qquad \partial^i$$
Also, is $$\dfrac{\partial}{\partial x^i} = \partial_i \quad \text{or}\quad \partial^i?$$
and why? What is the difference about the position of the indexes? I have troubles in understanding the difference of high and low index. We are working in $\mathbb{R}^3$ only, no higher dimensions.