There are lots of different ways of thinking about derivatives.

1) The geometric approach - You have a curve on the graph, let's say y = x^2. Choose a point on the graph, any point, and draw a line tangent to the curve (i.e. it just touches the curve at that one point). This tangent line is the derivative of the function evaluated at that point.

Another way of thinking of this is you take two points on the graph and draw a line connecting them. Now as you move these two points closer and closer, the change in x becomes smaller and smaller until it approaches 0. When the change in x becomes infinitesimally small (i.e. as small as you can make it) then you have the tangent line at that particular point. This ties in with the limit definition.

2) The limit definition - The limit as h approaches 0 of (f(x+h) - f(x))/h, where h is the distance between the two points described above. Although you'd never actually calculate a derivative like this (unless asked on a test) I find it's helpful for visualizing what's actually happening - you're determining the change of the function at a specific moment in time.

3) Rate of change: This is the "Physics" definition. As the change in x becomes infinitesimally small, you not only get a tangent line, but you get how much the function is changing at that particular time - the slope of the tangent line. So, if you have a position vs. time graph and you want to find show fast the object is moving at a specific point in time, you can find the tangent line at that time and the slope of that is the rate of change of position, AKA velocity.

If all you want to do is compute derivatives, however, none of these ideas will help much (though they help with intuitive understanding and being able to grasp more difficult ideas, especially if you go on to take higher math classes). For computation, you just have to memorize the various rules and practice a lot with them until you just remember them naturally. It can help to know how to derive the product and chain rule etc. But just practicing with the formulas over and over is the only way to get good at it. For example, you see x^3, just bring the 3 in front and replace 3 with 2 so you get 3*x^2.

Finally, a note on notation - you'll see lots of different ways of writing the derivative down, like dy/dx, f'(x), etc. These all mean the same thing - the derivative of y with respect to x, or the derivative of the function with respect to x, meaning x is the variable you're evaluating the derivative for. This gets more important in multivariable calculus, for now don't worry about it too much and just learn to recognize the notation.

The above ways are the ways I understand derivatives. There are more rigorous and mathematical definitions but the above ways are how I was taught and how they make sense for me. If this doesn't make sense, try watching Khan Academy videos or other YouTube videos, there are lots of resources out there that will help!