I have been learning calculus and I want to make sure I have it all right. From what I understand, the derivitive is like the slope of a function at a given point. The derivitive can be found by replacing x with (x + dx) and y with (y + dy), then solving for dy/dx. Integration gives you the function from the derivative, so integration is the opposite of differentiation. Please tell me if any of this is wrong. Also, I keep seeing a capital sigma sign, but I don't know what it is. Could someone tell me that also? I'm pretty sure it has something to do with calculus. Thanks.

The sigma sign is probably a summation sign. X = sum(a,b,c,d)
It is the same as an integral except it has no memory.
Given a curve S, the derivative is d(S) = S[j+1] - S[J] if dt is one.
Integral of S is i(S) = i(S) + S[j] or Total = Total + S[j]

Your x + dx and y + dy are fine.
Hope I haven't ruined calculus for you.

<i> The derivitive can be found by replacing x with (x + dx) and y with (y + dy), then solving for dy/dx.</i>

I find this statement confusing. From first principles, the derivative is defined to be:

f'(x) = lim(h->0) [ f(x+h) - f(x) ]/h

Here's an example. Suppose we want to find the derivative of the function f(x) = x². In that case:

f(x + h) = (x+h)² = x² + 2hx + h²
f(x) = x²

So f'(x) = lim(h->0) [(x² + 2hx + h²) - x²]/h

f'(x) = lim(h->0) [ 2hx + h²]/h

f'(x) = lim(h->0) [2x + h]

f'(x) = 2x

The sigma sign you keep seeing is nearly identical to the integral. You're probably encountering the Sigma notation with Riemann sums. With Riemann sums you're approximating the area of a function between 2 bounds by adding up the areas of many rectangles within the function. The base of the rectangles are always along the x-axis when calculating the area of a single function(or on some y line if you are to find the area between the function and the line y=?. The height is bounded by the top of the function, but that depends on whether you use a right hand, left hand or midpoint riemann sum. When you increase the number of rectangles you are increasing the accuracy of your approximation because the top of the rectangles are becoming more and more like the curve of the function as the number of rectangles increases. If you keep increasing the number of rectangles till it approaches infinity (the base of the rectangle will approach 0) you will get an incredibly accurate approximation of the area. Mathematicans gave it a special name, the Integral. Yup, the integral is practically nothing more than a riemann sum with an infinite amount of rectangles used to calculate area. Since you've been shown the awesomeness of integration you know that it is a great way to find area, later on you will be shown the beauty of volume, surface area, pressure, force and tons of other stuff like that with respect to integration. You'll probably learn other approximation methods such as Simpsons Rule, the Midpoint rule and the Trapezoidal rule. These are very useful when those cute little integration formulas and methods of integration fail. I hope you enjoy your journey through mathematics. Mine has just begun.