What are derivatives and integrals?

The_Z_Factor

What are they? In my book Im studying limits and it has mentioned a few times before and in the current chapter Derivatives and Integrals, but hasnt explained them. Could anybody explain what these two things are, exactly?

SiddharthM

if your asking for a formal definition then goto www.wikipedia.com and search for derivative and separately integration.

Geometric calculus interpretation:

If f(x) is a line and can be represented by mx +b then the slope is m, but this is a line and the slope is the same throughout the real numbers. Now consider y= x^2, what is the slope? it changes at each point, the slope at any point is the derivative of the function evaluated at that point.

the integral of a real function gives you the area under the curve of the function.

The_Z_Factor

So does this mean that there can be as many derivatives as there are points?

Gib Z

Different functions can be differentiated a different number of times.

Mute

It means that in general the derivative of a function of x is itself a function of x. i.e., the slope of a function is different at each point on that function.

For example, the derivative of x^2 is 2x. This means that on the curve y = x^2, at the point x = 4, the slope of the curve at x = 4 (or, perhaps more precisely, the slope of the line tangent to the curve at x = 4) is 2*4 = 8. Similarly, the slope at the point x = -5 is -10.

The_Z_Factor

Ah, thanks for clearing that up for me everybody, that explains it. I think as I'm beginning to learn more about simple calculus I'm beginning to like it more. Haha, it just might turn into a hobby once I learn enough.