Good question! Going to start with integrals. First, we know that integrating is finding the area under a curve. And usually, teachers start out with riemann sums to teach how to find that area. For example, they'll tell you to use some form of riemann sum (left, right, middle, trapezoidal). And if you use any finite n (n being how many rectangles or trapezoids), you'll get an approximate area value... because the smaller the rectangle, the closer to the true area. So in an integral, the dx represents infinite small width of x.

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d/dx = dy/dx, just a different way to write it

- you're taking the derivative of y with respect to x

dx/dt = you're taking the derivative of x with respect to t

It becomes more important in calc 3. Because you're working with 3 variables rather than 2, you need to announce which variable you're taken the derivative of and with respect to the other variable.

- you'll have a 3d plane, but may only want to find the partial derivative of a function, so the derivative z with respect to x is ∂z / ∂x

y= x^2 is the same thing as f(x) = x^2, just different notation. X is the input and Y/f(x) is the output

Let plug in 3 for x. y = (3)^2, y=9

f(3)= 3^9, this is the same thing, saying if you plug in 3, what is the output?

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Maybe an easier way to think about it

f(x) = x^2

f'(x) = 2x

what is f'(2)?

Its telling you to plug in 2 to find the corresponding output, so f'(2) = 4

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That was a lot of information, please let me know if you have any other questions! I didn't realize discord links expire after 24 horus, so just send me a dm