Actually, you can learn the basic ideas of calculus fairly easily. I mean in like 20 minutes. It's doing calculus, solving the problems, that kicks you in the ass.

So here goes, a basic bare-bones intro to calculus...

There are two big concepts in calculus: differentiation and integration. Differentiation is the process of finding out how much a function changes when you add just a teeny tiny bit to the x-value. In other words, if I add an almost microscopically small amount to x, how much does y change by? Doing this tells you the rate of change at an instantaneous point, i.e. instead of finding the average velocity between two points you can now find the instantaneous velocity at any single point in time.

If you picture a function's graph, then you can see that the entire graph consists of starting at some point and then moving to the right a tiny bit and finding the y value, plotting it, and then moving another infinitely-small step to the right, plotting it, etc etc. The end result is a continuous graph that was built by potentially millions of these tiny little steps. So integration is the process of taking all of those tiny steps and adding them all up, to find the area under the curve. Finding the area is hugely important, because for example in the graph where the derivative (the tiny step to the right) is the velocity, then integrating that graph gives you the total distance traveled. Which if you think about it makes sense, because the distance is the sum total of the velocity (the rate) multiplied by the time, which when plotted is an area.

Basic differentiation is easy, but most textbooks teach it in a way that confuses lots of new students. They are giving the correct rigorous method in the textbooks, but there's another approach, a better one IMO when first starting out, that makes 100% logical sense and it just works without blowing your brains out up front. Here's the really simple intuitive way to understand it.

If we have a function f(x) = x² and we want to make a tiny step to the right, then lets denote that tiny step as "dx" meaning "the differential of x" or "the tiny change in the x value". And we will denote the tiny change in y that results from this tiny change in x as "dy" aka "the differential of y".

So then we have this:

Original function: y = x²

Moving slightly to the right:

y + dy = (x + dx)²

Remember, this is "a tiny change in x" (adding dx) causes "a tiny change in y" (adding dy). Read "dx" as "dee-eks" and "dy" as "dee-why". dx is a single variable, not d multiplied by x! Same with dy!

Now, let's expand that out on the right:

y + dy = x² + 2x(dx) + (dx)²

(I'm using parens here to reinforce that dx is a single variable)

So now we have the original function x² , plus 2 tiny steps, plus a square of the tiny step.

But hold on, that tiny step is really small. Like, think of a flea on an elephant. It's tiny. So a really small number (a fraction, e.g. 1/x) squared is going to be reeeeeally small, a "second order of smallness". So now we aren't talking about a flea on an elephant, we are talking about a flea on the flea that is on the elephant. Think the elephant will notice? Nope.

So here's the magic of calculus. Just throw away that really tiny number. Because no matter how small it is, we can always make it smaller by just taking a smaller step to the right. So just throw it away.

(NOTE: This is not how it is taught in modern texts, but is how it was taught in the old days. Modern texts teach a very rigorous method that proves you can essentially throw that number away because it can be proven to equal zero. In fact that's the first thing you are taught. But that's too deep for learning the basics, so go with the old guys, its way more intuitive and fun.)

So now, throwing out the dx² we now have:

y + dy = x² + 2xdx

Important note

What is the value of y? Well, we said it was x² because that is the function, right? So y = x^2. So, by the power of algebra, subtract x² from both sides and we have:

dy = 2xdx

You can leave it in this form, or more commonly divide both sides by the dx to show the ratio:

dy/dx = 2x

And tada, ladies and gentlemen, you have just derived a new function (the derivative) that, when given any x-value, will tell you the instantaneous rate of change (instantaneous slope at a single point) of the original function!

Now, give the new function a name, f'(x), read "f-prime of x".

So, remember f(x) = x² and f'(x) = 2x...

At x = 1, f(x) = 1, and the slope at point (1, 2) is f'(1) = 2(1) = 2!

At x = 2, f(x) = 4, and the slope at point (2, 4) is f'(2) = 4!

At x = 3, f(x) = 9, and the slope at point (3, 9) is f'(3) = 6!

So if we have an asteroid coming at Earth and its position over time is modeled by f(x) = x² (wow!) then you can now find its velocity at any instant, and project out exactly where it will be at some arbitrary time in the future. Which means you can figure out when and where it will hit, and hide accordingly!

There's so much more, but this is to me the fun way to learn it. Learn the rigorous stuff later, start with the fun stuff. :)

NB if you like the way this is approached, it is largely stolen from the century-old book Calculus Made Easy which you can read an old edition of here. As the author said, "What one fool can do, another can."