Understanding Differentiation

Discussion in 'Math' started by Barnaby Walters, Jun 16, 2011.

  1. Barnaby Walters

    Thread Starter Member

    Mar 2, 2011
    103
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    Hello all,

    Background: I finished GCSE maths last year with an A, and I'm revising for GCSE stats at the moment — my exam's next week. I have no intention of taking any more academic maths tests, but I want to expand my understanding, both for electronics and to make my brain work in different ways.

    I'm trying to understand some 'higher level' maths (Although I know that this stuff will be mind-bogglingly simple to many here), starting with basic calculus. I just want to check that my current understanding is just about right:

    Differentiation is the process of taking an equation (known as a function?) and finding the function that gives the original function's gradient at any given point. There are rules, such as dy/dx of x^n = nx^n-1. Like a function that takes a numerical input and gives a numerical output depending on rules, differentiation is a function that takes a function as input and 'processes' it, giving another function as an output.

    Does that sound just about right?

    Thanks,
    Barnaby
     
  2. Georacer

    Moderator

    Nov 25, 2009
    5,142
    1,266
    As you said, just about right. Differentiation is a process that takes a function and gives another.
    You can also think of the derivative of a function as a function that shows "how fast" that function increases or decreases at any gives point.

    I 'd like to point out that there's a difference between a function and an equation. A function f(x) is an expression, a formula if you like, that takes a value of x and builds with it another value y. When we write y=f(x) we just define y as a function of x. It's not a proper equation.
    On the other hand, when you write f(x)=0, then you have an equation in your hands. One with an expression of x in one end and 0 in the other.
     
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  3. steveb

    Senior Member

    Jul 3, 2008
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    I also agree that it is just about right.

    One point is that the term gradient, used to describe differentiation, is fine, but remember when dealing with functions of one variable, such as y=f(x) the operator generates a function from a function. However, in general the gradient is a vector, which implies a more advanced "operator". The gradient will, in general, take a scalar function and give a vector function, which is actually a collection of many functions. This is a higher level math issue which does not need to be brought up, but I mention it because you used the word gradient which has a specific math meaning aside from the colloquial usage. It might be better to use words like "slope" and "rate of change" in simplified discussions.

    Also, it's good to point out that differentiation is an operation which takes a limit. Even muliplying by two is an operation that takes a function and spits out another function, but differentiation is the limit of a function of two variables. That is it is the limit of g(x, \Deltax) , as \Deltax goes to zero, where g(x,\Deltax)=(f(x+\Deltax)-f(x))/\Deltax.
     
    Last edited: Jun 16, 2011
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  4. Barnaby Walters

    Thread Starter Member

    Mar 2, 2011
    103
    4
    Hi there,

    Fab, thanks for confirming. It's taken me a couple of weeks and several different resources to get this far.

    So, this is quite similar to programming — even the same terminology and syntax!

    Okay, there were lots of phrases in there which I've never heard before! I am vaguely familiar with vectors as ways of describing movement, but have not come across vector functions. Are they functions that take vectors as their arguments?

    Right — limits are what I'm moving on to try to figure out yet. From wikipedia I take it that limits have replaced infinitesimals, neither of which I have much of a clue about :) So, that's my next few weeks of informal study!

    Thanks for your help,
    Barnaby
     
  5. studiot

    AAC Fanatic!

    Nov 9, 2007
    5,005
    513
    Hello neighbour,

    The idea of a limit is very simple, although sometimes it can be quite difficult to work out the value of a particular limit.

    I just had a sneak peek at Wikipedia and they make it very complicated so try this.

    Taking a limit is a process not a formula.

    Take a half.
    The distance between one half and one is a half; take half of this ie a quarter and add to the original half.

    So we have

    \frac{1}{2} + \frac{1}{4} = \frac{3}{4}

    Take a half of the distance between this and one ie one eigth and add

    \frac{1}{2} + \frac{1}{4} + \frac{1}{8} +  = \frac{7}{8}

    Keep going until you get bored

    \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{{16}} = \frac{{15}}{{16}}

    You can see that each time we are getting closer and closer to 1.

    We call 1 the 'limit' of this process. We can get as close as we like to the limit, but we only ever reach it by 'going to infinity' .

    Most of the rest of the theory of limits is about clever ways of avoiding slogging through all this process, especially where the limit is not so obvious.

    Hope this is useful.
     
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  6. steveb

    Senior Member

    Jul 3, 2008
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    So, don't worry too much if I mention something unfamiliar. I was just cautioning that the term gradient, which I think you meant in the general sense, also has a specific mathematical meaning. However, your usage was perfectly fine, as long as we understand you don't mean the formal mathematical gradient, which is a more particular type of derivative.

    So generally, you can have vector functions that have vectors as both inputs and outputs. However, the gradient takes one scalar function as an input and generates 1, 2, 3, 4 or more functions (depending on the number of dimensions of the space) that form the components of a vector (or more precisely a special type of vector called a one-form, if you want even more confusing terminology). This is all several steps down the road for you, so don't worry too much about it yet. You are correct to focus on limits now as the next step, and you also seem well-prepared to do that.
     
  7. Barnaby Walters

    Thread Starter Member

    Mar 2, 2011
    103
    4
    Studiot — thanks, that's an excellent explanation of limits! I can see roughly what they are now. Wikipedia does tend to be over-technical, but I have found that wikibooks sometimes has more structured, friendly introductions to things like calculus (Calculus one is at: http://en.wikibooks.org/wiki/Calculus)

    Thanks,
    Barnaby
     
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