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Math Help - Differentials

  1. #1
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    Differentials

    edit - solved...haha i just realised you differentiate with respect to one first, then add the differential with respect to the other.

    Hi everyone,

    I just need some help on a question that I'm currently trying to solve. I'm doing Summer Maths at uni at the moment but I haven't quite grasped this concept.

    The question is telling me to calculate a differential, given an equation.

    A = xy

    Find dA.


    I know for a fact that the answer is dA = ydx + xdy, but I am unsure as to how the lecturer got to that process because he stated it was 'fairly self explanatory'. I guess I didn't have the guts to ask infront of a hundred people so here I am...haha

    I don't expect a complete answer...moreso some guidance or references that will help me along the way.

    Thanks in advanced,
    Last edited by ivanyo; December 28th 2011 at 09:20 PM. Reason: solved
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  2. #2
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    Re: [SOLVED]Differentials

    Quote Originally Posted by ivanyo View Post
    edit - solved...haha i just realised you differentiate with respect to one first, then add the differential with respect to the other.

    Hi everyone,

    I just need some help on a question that I'm currently trying to solve. I'm doing Summer Maths at uni at the moment but I haven't quite grasped this concept.

    The question is telling me to calculate a differential, given an equation.

    A = xy

    Find dA.


    I know for a fact that the answer is dA = ydx + xdy, but I am unsure as to how the lecturer got to that process because he stated it was 'fairly self explanatory'. I guess I didn't have the guts to ask infront of a hundred people so here I am...haha

    I don't expect a complete answer...moreso some guidance or references that will help me along the way.

    Thanks in advanced,
    Differentiate both sides with respect to x. Since y is a function of x, on the RHS you need to use the product rule...
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  3. #3
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    Re: Differentials

    Suppose x and y are functions of some paramter, t: x= x(t), y= y(t). Then we can think of f(x,y) as a function of the single variable t.

    By the chain rule, \frac{df}{dt}= \frac{\partial f}{\partial x}\frac{dx}{dt}+ \frac{\partial f}{\partial y}\frac{dy}{dt}.

    The differential, for a function of a single variable, is defined by df= \frac{df}{dt}dt so we have here
    df= \frac{df}{dx}dt= \left(\frac{\partial f}{\partial x}\frac{dx}{dt}+ \frac{\partial f}{\partial y}\frac{dy}{dt}\right)dt
    = \frac{\partial f}{\partial x}\left(\frac{dx}{dt}dt\right)+ \frac{\partial f}{\partial y}\left(\frac{dy}{dt}dt\right)= \frac{\partial f}{\partial x}{dx}+ \frac{\partial f}{\partial y}dy
    which is now independent of t.
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