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Thread: Challening Differentiatioin Problem

  1. #1
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    Question Challening Differentiatioin Problem


    If :
    z=x+2y

    While:
    y=r sinθ
    x= r cosθ


    Find :-

    1- (dz/dx)y
    2- (dz/dx)r
    3- (dz/dx)θ

    4- (dz/dy)x
    5- (dz/dy)r
    6- (dz/dy)θ

    7- (dz/dθ)x
    8- (dz/dθ)y
    9- (dz/dθ)r

    10- (dz/dr)θ
    11- (dz/dr)x
    12- (dz/dr)y

    13- dz/dxdy
    14- dz/dxdθ
    15- dz/dydθ

    16- dz/drdx
    17- dz/drdθ
    18- dz/dxdy

  2. #2
    Elite Member stapel's Avatar
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    Cool

    Quote Originally Posted by Simo View Post
    If : z=x+2y

    While:
    y=r sinθ
    x= r cosθ

    Find :-

    1- (dz/dx)y
    2- (dz/dx)r
    3- (dz/dx)θ

    4- (dz/dy)x
    5- (dz/dy)r
    6- (dz/dy)θ

    7- (dz/dθ)x
    8- (dz/dθ)y
    9- (dz/dθ)r

    10- (dz/dr)θ
    11- (dz/dr)x
    12- (dz/dr)y

    13- dz/dxdy
    14- dz/dxdθ
    15- dz/dydθ

    16- dz/drdx
    17- dz/drdθ
    18- dz/dxdy
    Where are you stuck? Please be complete. Thank you!


  3. #3
    Full Member
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    Hello Simo:

    Please take time to read the forum guidelines; you can start with this link to the summary page.

    Cheers

  4. #4
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    I presume that the letter after the derivative indicates that the derivative is to be taken holding that variable constant.

    So here is the idea for the first few:

    1) [tex]\left(\frac{\partial z}{\partial x}\right)_y[/tex]

    Since [tex]z= x^2+ 2y^2[/tex]. Holding y constant, the derivative with respect to x is 2x.

    2) [tex]\left(\frac{\partial z}{\partial x}\right)_r[/tex]
    [tex]x^2+ y^2= r^2cos^2(\theta)+ r^2 sin^2(\theta)= r^2(cos^2(\theta)+ sin^2(\theta))= r^2[/tex] so [tex]y^2= r^2- x^2[/tex]
    [tex]x^2+ 2y^2= x^2+ 2(r^2- x^2)= -x^2+ 2r^2[/tex]. Holding r constant, the derivative with respect to x is -2x.

    3) [tex]\left(\frac{\partial z}{\partial x}\right)_\theta[/tex]
    [tex]\frac{y}{x}= \frac{r sin(\theta)}{r cos(\theta)}= tan(\theta)[/tex] so [tex]y= x tan(\theta)[/tex] and [tex]x^2+ 2y^2= x^2+ 2x^2 tan^2(\theta)= x^2(1+ 2 tan^2(\theta))[/tex]. Holding [tex]\theta[/tex] constant the derivative with respect to x is [tex]2x(1+ 2tan^2(\theta))[/tex]

    Now, YOU try the others!
    Last edited by HallsofIvy; 05-12-2014 at 08:37 PM.

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