Derivatives can be understood in another way by using infinitesimal calculus, which relies on algebra and infinitesimals, rather than limits, to define calculus. It's not as easy to extend as limit calculus, but it's more intuitive, and the results are the same.
Suppose you have an equation, such as y = x². If you increase x by a minute amount, what is the increase on y? To determine that increase, you use calculus.
Let's call the minute increase on y dy, and the minute increase on x dx. So:
y + dy = (x + dx)²
Multiply out (x + dx)² to get:
y + dy = x² + 2x*dx + (dx)²
Now, we defined dx as being minutely small. Much like how a second is minute compared to a minute, it is even more minute compared to an hour. So (dx)² is so small, we can disregard it.
y + dy = x² + 2x*dx
I know, that sounds like real bullshit, but bear with me.
The original equation was y = x². Subtract that out.
dy = 2x*dx
Now, we divide by dx. (Again, sounds like bullshit, but...)
dy/dx = 2x
And there we go. The derivative of x² is 2x. When you increase x² by a certain amount, it is increased by 2x. Similarly with decreases.
A derivative is the value by which change is determined.