How can I get unblocked on learning Calculus? I love math and science. In fact paid for $1/2$ my college tuition by tutoring algebra and trigonometry. But when it came to calculus, I became blocked. I understand the concepts of speed and rate of change, but when I start seeing all the symbols like $f'(x)$ and $dx$, my mind goes all fuzzy.
My learning style is that if I can picture something and comprehend it in real life, then I get it in symbols. Algebra, geometry, and trig are all easy enough to picture. But what does it look like in real life to 'take the derivative' or the integral of something? Does it look like a sphere becoming a circle?
Does anyone else's brain work like this? How did you 'get' calculus?
 A: Im going to share with you my (short) experience regarding mathematics. It will not be a direct answer to your question but maybe it will help you! 
So when I was in school, I loved maths. I was actually good at them. Trigonometry, algebra and even pre-calculus. I even thought that was all what maths were about. However, when 4 years ago, I entered the university, I started realizing maths wasn't even about numbers (in many cases). I suddenly found my self dealing with concepts that were absolutely imposible to picture them somehow. Not only that, suddenly I was solving a problem that I would have never thought that it could even be called maths. This made me feel seriously mixed up. When I decided to study mathematics, I am sure I wasn't $picturing$ my self doing what I am doing today. However, I am still studying, and every time, more eagerly. And that's the important thing I want to share with you. Don't think about maths as something you can picture in reality. That will only stop you. Actually, the great thing about maths, which makes it $ perfect $ is that it doesn't have to even explain reality!
I like to think of maths as this indescriptible magic stuff that can make you love it or hate it. And if you do love it, just learn from it, don't look for any reality in it. You have physics for that!
A: If you want to "picture" something in calculus, here's one example:


*

*The size of a boundary times the rate at which the boundary moves, equals the rate of change of size of the bounded region.


(Privately I think of this as "the boundary rule".  Does anyone know a better name for it?)
Examples:


*

*A region is enclosed by a sphere.  The change in the radius is how far the boundary moves.  Hence the rate at which the boundary moves is the rate of change of the radius.  The size of the boundary is the surface area of the sphere.  Multiply those to get the rate at which the volume changes.  (This explains why the formula for the surface area of a sphere as a function of the radius is the derivative with respect to the radius, of the volume of the sphere.)

*An $n$-cube has sides of length $x$, one corner right at the origin, and $n$ edges on the coordinate axes.  As $x$ changes, $n$ of the faces move, and they move at the rate at which $x$ changes.  Each of them has size $x^{n-1}$.  So the rate of change of volume, i.e. the rate of change of $x^n$, is $$\underbrace{x^{n-1}+\cdots+x^{n-1}}_{n\text{ terms}}$$ times the rate at which $x$ changes.
A: I can understand why your mind goes fuzzy when you see things like $x^2 dx$ and $\frac{dy}{dx}$, because these are tricky concepts that are difficult to formalize and frequently used incorrectly, and their geometric meaning is pretty complicated. In fact, even after 5 or so years of university mathematics, I honestly still don't get the geometric meaning of $\frac{dy}{dx}$. However, the meaning of the notation $f'(x)$ should be perfectly clear, and you should be able to visualize it with only a little bit of thought. If you can't do so at the moment, perhaps the problem is that maybe you are missing is the concept of a function. I won't try to explain it here, but I suggest searching online for an explanation of the function concept. Some key words:


*

*Set, function

*Domain, codomain; make sure you know what the notation $f : X \rightarrow Y$ means, where $X$ and $Y$ denote sets.

*Injective (one-to-one), surjective (onto)

*Higher-order function

*Lambda abstraction


Another important point is that, given a function $f : \mathbb{R} \rightarrow \mathbb{R},$


*

*$f$ can usually be visualized as a curve in the plane.

*the notation $f(x)$ ("$f$ evaluated at $x$") can be visualized as the height of $f$ at $x$.


Make sure you understand this.
Once you've got these concepts, calculus shouldn't make your mind go numb anymore. A simple way to understand the notation $f'(x)$ is that it really means the slope of $f$ at $x$. In other words, its the derivative of $f$, evaluated at $x$. Try to think of derivatives as higher-order functions; the notation $f'(x)$ really means something more like $$(D(f))(x),$$ where $D$ is the derivative function (which is higher-order). The expression $(D(f))(x)$ means: start with the function $f$, then apply $D$ to it, thereby obtaining the corresponding "slope function" $D(f)$, and then evaluate this new function $D(f)$ at $x$; you can visualize this as the height of the slope function $D(f)$ at $x$.
A: The concepts you mention were historically derived from geometrical and physical problems, like curves and areas, motion etc.
You can get them explained very close to everyday conception.
And we might try this here if you break it down in single questions.
However the real power of mathematics is to abstract, leaving out everything except the essential properties. This allows applications to a broader range. 
E.g. you can work out geometry in 10 dimensions, which comes in handy if you have an optimization problem in 10 sorts of different products.
Also everyday conception can be misleading, that is why later calculus got more abstract methods to achieve more rigour.
So you might use conception to get an initial grasp of the subject, but the step to abstraction can not be fully avoided if you want to do modern mathematics.
