Yes. That is the true definition of derivative.

20 = 10(?)
You can't compare like that la, bro. f(x) = (x*x) = x+x+...x+x is flawed the "...." was not define. Also, does that mean:
d/dx (x^3) = d/dx(x+x+....x+x) (x^2 number of times)
d/dx (x^3) = 1+1+....1+1

if x=10;
d/dx (x^3) = 1+1+....1+1 (10^2 number of times)
d/dx (x^3) = 100

but, d/dx(x^3) = 3x^2 = 300

Pardon the nitpicking:

Actually, that isn't the 'true' definition of the derivative. You've provided the notion of what the derivative is, namely that it is the rate of change of some function with respect to some variable at some certain value of that variable.

Also, the '...' was actually defined. It had already been stated that the sum occurs x times (yes it's defined, even if x is not a positive natural number).

But otherwise you're mostly right, nailing down that the problem is we've not considered the fact that the x's are summed x times, which is in itself a variable

From wikipedia:


In school, to avoid confusion, it is often express as the slope of a graph. In fact, some math teachers could hardly appreciate the rate of change. The development of calculus, derivative and integration was largely due to extensive study of transient (the rate of change and behavior) of different physic aspect.

What you quoted agrees with what I said, that informally, or loosely speaking, the derivative is the rate of change of some function, or equivalently, the notion of the derivative. It is not, however, the 'true' definition of a derivative, which is what you stated. You'd need limits to construct the definition of the derivative.

This post has been edited by bgeh: Aug 1 2009, 04:52 PM