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Thread: Differentiation

  1. #1 Differentiation 
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    Can anyone teach me how to differentiate this one?y=x(1-3x)^5


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    Forum Sophomore Matt Lacey's Avatar
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    Ooh, it's been a while since I've done any differentiation, but I'll give it a go.

    This differentiation will require a combination of the product rule and the chain rule, so:

    y = x(1-3x)<sup>5</sup>

    Suppose u = x and v = (1-3x)<sup>5</sup>

    y' = uv' + u'v

    u' = 1, that's easy. v' is a bit more complicated, and requires the chain rule.

    v = (1-3x)<sup>5</sup>

    Suppose that v = w<sup>5</sup> and w = (1-3x)

    dv/dx = dw/dx x dv/dw

    dw/dx = -3 and dv/dw = 5u<sup>4</sup>

    Hence:

    v' = -3(5(1-3x)<sup>4</sup>) = -15(1-3x)<sup>4</sup>

    Put that into the original expression:

    y' = uv' + u'v = x(-15(1-3x)<sup>4</sup> + (1-3x)<sup>5</sup>

    y' = (1-3x)<sup>5</sup> - 15x(1-3x)<sup>4</sup>

    Probably simplifies to something a lot nicer, but I can't really be bothered at the moment, sorry :P I *think* I've done it right, but it's been a while so I can't really be sure anymore


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    Forum Professor river_rat's Avatar
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    Got the same thing Matt

    It simplifies to (1 - 3 x)<sup>4</sup> (1 - 18 x)
    As is often the case with technical subjects we are presented with an unfortunate choice: an explanation that is accurate but incomprehensible, or comprehensible but wrong.
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  5. #4  
    Forum Sophomore Matt Lacey's Avatar
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    Yeah I got that simplification too. Couldn't be bothered to edit the post though, I didn't consider it an amazing improvement, haha.
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  6. #5  
    Forum Professor river_rat's Avatar
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    I tutor engineers this stuff so the simplification comes naturally.
    As is often the case with technical subjects we are presented with an unfortunate choice: an explanation that is accurate but incomprehensible, or comprehensible but wrong.
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  7. #6  
    Forum Sophomore Matt Lacey's Avatar
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    Cool. This forum gives me the odd good excuse to keep up on my maths Since I'm a materials chemist these days I don't have much use for pure maths anymore :P
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  8. #7  
    Forum Professor river_rat's Avatar
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    So you're a lab jockey - not for me!

    Welcome to the forum
    As is often the case with technical subjects we are presented with an unfortunate choice: an explanation that is accurate but incomprehensible, or comprehensible but wrong.
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  9. #8  
    Forum Sophomore Matt Lacey's Avatar
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    Thanks I'm starting to like this place
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  10. #9  
    Forum Professor wallaby's Avatar
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    Quote Originally Posted by Matt Lacey
    dv/dx = dw/dx x dv/dw

    dw/dx = -3 and dv/dw = 5u<sup>4</sup>
    how does dv/dw become 5u<sup>4</sup>

    what is 'u' anyhow?
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  11. #10  
    Forum Professor river_rat's Avatar
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    Its a typo wallaby

    Should read

    dv/dx = dw/dx x dv/dw

    dw/dx = -3 and dv/dw = 5w<sup>4</sup>

    where w = (1 - 3x)
    As is often the case with technical subjects we are presented with an unfortunate choice: an explanation that is accurate but incomprehensible, or comprehensible but wrong.
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  12. #11  
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    Use the formulae d(u*v)=u*d(v)+v*d(u) and d(x^n)=n*x^(n-1), n is constant in your question and simplify to get final result. Afterall, its your homework and I need not do intermediate steps for you
    K. Srinivasa Ramanujam
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  13. #12  
    Forum Sophomore Matt Lacey's Avatar
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    Quote Originally Posted by river_rat
    Its a typo wallaby

    Should read

    dv/dx = dw/dx x dv/dw

    dw/dx = -3 and dv/dw = 5w<sup>4</sup>

    where w = (1 - 3x)
    Sorry, my bad

    I had it stuck in my head to use dy/dx = du/dx x dy/du, temporarily forgot I was using different symbols. Oooops...
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  14. #13  
    Forum Professor wallaby's Avatar
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    Quote Originally Posted by Matt Lacey
    Quote Originally Posted by river_rat
    Its a typo wallaby

    Should read

    dv/dx = dw/dx x dv/dw

    dw/dx = -3 and dv/dw = 5w<sup>4</sup>

    where w = (1 - 3x)
    Sorry, my bad

    I had it stuck in my head to use dy/dx = du/dx x dy/du, temporarily forgot I was using different symbols. Oooops...
    thanks for clearing that up, i myself am new to differentiation so it is easy for me to get confused.
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  15. #14  
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    can anyone tell me what are the uses of differentiation?>
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  16. #15  
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    You bastard Matt - beat me to it :-)
    If a man has a why, he can bear almost any how. - Nietzsche
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  17. #16  
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    Quote Originally Posted by cheenaik
    can anyone tell me what are the uses of differentiation?>
    The differential of a function gives you a formula for the gradient of that function at any x (or whatever the independent variables are).

    so, if y = x^2, dy/dx = 2x which means at x=3 the gradient of x^2 is 6.

    In general, differentation can be described as the 'rate of change of something with respect to something else'.

    I'm sure others can give a slighter more rigorous reply
    If a man has a why, he can bear almost any how. - Nietzsche
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  18. #17  
    Forum Professor river_rat's Avatar
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    Quote Originally Posted by cheenaik
    can anyone tell me what are the uses of differentiation?>
    Differentiation is important as is
    • Gives possible local maximum and minimimum values for a "nice" function (optimisation and mathematical programming problems)
    • Is the definition for velocity and acceleration so forms the basis for dynamics and kinematics
    • Derivatives are used to give approximations to rather nasty looking functions (ie. Taylor expansions)
    • Derivitives are used to solve equations numerically (ie Newtons method)
    • Most importantly, as derivitives give "rate of change" information they form the basic ingrediants for Differential Equations which are ubiquitous in the sciences and one of the main friving forces for modern mathematical research.


    Thats just off the top of my head, there are a few more reasons - anybody got anything else to add?
    As is often the case with technical subjects we are presented with an unfortunate choice: an explanation that is accurate but incomprehensible, or comprehensible but wrong.
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  19. #18 Re: Differentiation 
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    Quote Originally Posted by cheenaik
    Can anyone teach me how to differentiate this one?y=x(1-3x)^5
    The equation is

    You can use what's known as the "chain rule" to differentiate this function. Firstly however, you must learn a less sophisticated rule known as the "power rule."

    Assume we have a function , in order to find the derivative of it, we must plug it into the the difference quotient, which is the following formula:

    . From this, we derive the following formula (note that we can't pass to limit yet, as our output is an undefined quantity, therefore we must make some cancellations first):



    Now, we can use the binomial theorem to simplify the numerator to the form , where are the remaining terms within the expansion of the binomial. Now, we can make our cancellations: and to begin with cancel one another out.

    From that, we have the remaining expression:



    Based on the properties of reciporicals, the above expression simplifies to

    Now we can simply pass to the limit, or , which is based on the fact that all of the remaining terms have a coefficient of and the change in x tends to zero. What this means is that .

    Here is where the chain rule comes into play: it states that if we have a compound function , than its derivative is

    Than the derivative of the function is equal to
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