What is differentiation?

Awwh that's so cute, you actually think you're smart just because you can throw some more technical words in to confuse the OP!

He never even asked about keeping it real valued. Maybe if he had it'd give him something a bit more to look at, but it's actually completely useless in every way. Every other answer, that just tries to be helpful instead, works regardless of where you're applying it. So really, I would love to be wrong, what were you trying to do here...

What are all the d's and what's lim?
Apart from knowing that it finds the gradient of a curve, i haven't a clue what it is.

It seems like Maths is not for me, because I just don't get the whole idea of defferentiation and Intergration. In my opinion both are useless and will not help me in the real world.

The derivative of a real-valued function $f$ in a domain $D$ is the Lagrangian section of the cotangent bundle $T^*(D)$ that gives the connection form for the unique flat connection on the trivial $\boldsymbol R$-bundle $D \times \boldsymbol R$ for which the graph of $f$ is parallel.

Is this straight out of 'Maths made Difficult'?

I've heard of that book! I've never found a copy to read though, and it seems to be out of print...

Same here, I've searched pretty thoroughly for it and all I've found is an illicit pdf of part of it and a ridiculously expensive second hand copy on Amazon.

I've heard of that book! I've never found a copy to read though, and it seems to be out of print...

Lim describes what happens to a function, f(x) as x gets close to a particular value eg:

$\displaystyle{\lim_{x->12} (x+1)} = 13$ because as x gets closer to 12, x+1 gets closer to 13.

$\displaystyle{\lim_{x->2} \frac{1}{x-2}} = \infty$ because as x gets close to 2, 1/(x-2) approaches infinity.

This one's a bit more interesting
$\displaystyle{\lim_{x->0} \frac{\sin x}{x}} = 1$

think he was looking for a simpler answer

Are people who are good at maths clever human beings?

What are all the d's and what's lim?
Apart from knowing that it finds the gradient of a curve, i haven't a clue what it is.

Differential calculus is the study of finding the instantaneous slope of a curve, not merely the slope between two points but the slope at a single point. The function that shows the slope at every point is called the derivative of the curve. Usually the instantaneous slope (at each particular point) is defined as the limit of slopes as the second point gets closer to the first. If you work in a framework with infinitesimals, however, the instantaneous slope can also be computed as the slope between two infinitesimally close points. For instance, in smooth infinitesimal analysis, the basic axiom is that

f(x+d) = f(x) + f'(x)d

for any nilsquare d^2 = 0; this may be where your d's come from. In standard calculus, the notion of d's are basically the cultural artifacts that remain from this kind of reasoning.

I'm pretty sure Newton and Leibniz didn't use this sort of reasoning, and if they did, they didn't use this terminology... Moreover, to use this sort of analysis, you need to first extend the domain of your functions to include these infinitesimals. This, I think, is a non-trivial task. It's obvious enough how to do it for polynomials and power series, but what if I give you an arbitrary continuous, differentiable function?

Differential calculus is the study of finding the instantaneous slope of a curve, not merely the slope between two points but the slope at a single point. The function that shows the slope at every point is called the derivative of the curve. Usually the instantaneous slope (at each particular point) is defined as the limit of slopes as the second point gets closer to the first. If you work in a framework with infinitesimals, however, the instantaneous slope can also be computed as the slope between two infinitesimally close points. For instance, in smooth infinitesimal analysis, the basic axiom is that

f(x+d) = f(x) + f'(x)d

for any nilsquare d^2 = 0; this may be where your d's come from. In standard calculus, the notion of d's are basically the cultural artifacts that remain from this kind of reasoning.

What on Earth is this post all about? Of course good mathematicians are clever people...

Lim describes what happens to a function, f(x) as x gets close to a particular value eg:

$\displaystyle{\lim_{x->12} (x+1)} = 13$ because as x gets closer to 12, x+1 gets closer to 13.

$\displaystyle{\lim_{x->2} \frac{1}{x-2}} = \infty$ because as x gets close to 2, 1/(x-2) approaches infinity.

This one's a bit more interesting
$\displaystyle{\lim_{x->0} \frac{\sin x}{x}} = 1$