Derivative of a multivariable function at a point [closed]

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I'm facing this problem of finding the derivative of a function $f(x,y)= (sin^2 x cdot cos y, xy)$ at the point $(pi,pi/2).$ The problem is that I don't know if I should calculate the partial derivative and then plug in the points or find the directional derivative or what exactly because it's the first time to find such function with two components.




multivariable-calculus derivatives partial-derivative




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closed as off-topic by José Carlos Santos, kjetil b halvorsen, user 108128, amWhy, user223391 Jul 31 at 1:07


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  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – José Carlos Santos, kjetil b halvorsen, user 108128, amWhy, Community
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    I think you could find very useful this question math.stackexchange.com/questions/195000/…
    – Davide Morgante
    Jul 30 at 19:37










  • Thanks, it was very useful.
    – Mohamed Mossad
    Jul 30 at 21:05














up vote
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down vote

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I'm facing this problem of finding the derivative of a function $f(x,y)= (sin^2 x cdot cos y, xy)$ at the point $(pi,pi/2).$ The problem is that I don't know if I should calculate the partial derivative and then plug in the points or find the directional derivative or what exactly because it's the first time to find such function with two components.




multivariable-calculus derivatives partial-derivative




share|cite|improve this question













closed as off-topic by José Carlos Santos, kjetil b halvorsen, user 108128, amWhy, user223391 Jul 31 at 1:07


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – José Carlos Santos, kjetil b halvorsen, user 108128, amWhy, Community
If this question can be reworded to fit the rules in the help center, please edit the question.








  • 1




    I think you could find very useful this question math.stackexchange.com/questions/195000/…
    – Davide Morgante
    Jul 30 at 19:37










  • Thanks, it was very useful.
    – Mohamed Mossad
    Jul 30 at 21:05












up vote
0
down vote

favorite









up vote
0
down vote

favorite











I'm facing this problem of finding the derivative of a function $f(x,y)= (sin^2 x cdot cos y, xy)$ at the point $(pi,pi/2).$ The problem is that I don't know if I should calculate the partial derivative and then plug in the points or find the directional derivative or what exactly because it's the first time to find such function with two components.




multivariable-calculus derivatives partial-derivative




share|cite|improve this question













I'm facing this problem of finding the derivative of a function $f(x,y)= (sin^2 x cdot cos y, xy)$ at the point $(pi,pi/2).$ The problem is that I don't know if I should calculate the partial derivative and then plug in the points or find the directional derivative or what exactly because it's the first time to find such function with two components.





multivariable-calculus derivatives partial-derivative





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share|cite|improve this question




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edited Jul 30 at 23:06









Michael Hardy

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204k23185460









asked Jul 30 at 19:32









Mohamed Mossad

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102




closed as off-topic by José Carlos Santos, kjetil b halvorsen, user 108128, amWhy, user223391 Jul 31 at 1:07


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – José Carlos Santos, kjetil b halvorsen, user 108128, amWhy, Community
If this question can be reworded to fit the rules in the help center, please edit the question.




closed as off-topic by José Carlos Santos, kjetil b halvorsen, user 108128, amWhy, user223391 Jul 31 at 1:07


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – José Carlos Santos, kjetil b halvorsen, user 108128, amWhy, Community
If this question can be reworded to fit the rules in the help center, please edit the question.







  • 1




    I think you could find very useful this question math.stackexchange.com/questions/195000/…
    – Davide Morgante
    Jul 30 at 19:37










  • Thanks, it was very useful.
    – Mohamed Mossad
    Jul 30 at 21:05












  • 1




    I think you could find very useful this question math.stackexchange.com/questions/195000/…
    – Davide Morgante
    Jul 30 at 19:37










  • Thanks, it was very useful.
    – Mohamed Mossad
    Jul 30 at 21:05







1




1




I think you could find very useful this question math.stackexchange.com/questions/195000/…
– Davide Morgante
Jul 30 at 19:37




I think you could find very useful this question math.stackexchange.com/questions/195000/…
– Davide Morgante
Jul 30 at 19:37












Thanks, it was very useful.
– Mohamed Mossad
Jul 30 at 21:05




Thanks, it was very useful.
– Mohamed Mossad
Jul 30 at 21:05










1 Answer
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You can calculate the partial derivatives, and see that they`re continous. This means the function is differentiable .



$$f_1(x,y)=sin^2(x)cos(y)$$



$$f_2(x,y)=xy$$



$frac partial f_1 partial x=cos(y)2sin(x)cos(x), frac partial f_1 partial y=-sin^2(x)sin(y)$



$frac partial f_2 partial x=y, frac partial f_2 partial y=x$.



So the derivative is a $2x2$ matrix $D_f(v)=frac partial f_i partial x_j$. (Use $x_1=x$ and $x_2=y$ for simplicity of writing)






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    1 Answer
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    active

    oldest

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    1 Answer
    1






    active

    oldest

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    active

    oldest

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    active

    oldest

    votes








    up vote
    0
    down vote



    accepted










    You can calculate the partial derivatives, and see that they`re continous. This means the function is differentiable .



    $$f_1(x,y)=sin^2(x)cos(y)$$



    $$f_2(x,y)=xy$$



    $frac partial f_1 partial x=cos(y)2sin(x)cos(x), frac partial f_1 partial y=-sin^2(x)sin(y)$



    $frac partial f_2 partial x=y, frac partial f_2 partial y=x$.



    So the derivative is a $2x2$ matrix $D_f(v)=frac partial f_i partial x_j$. (Use $x_1=x$ and $x_2=y$ for simplicity of writing)






    share|cite|improve this answer

























      up vote
      0
      down vote



      accepted










      You can calculate the partial derivatives, and see that they`re continous. This means the function is differentiable .



      $$f_1(x,y)=sin^2(x)cos(y)$$



      $$f_2(x,y)=xy$$



      $frac partial f_1 partial x=cos(y)2sin(x)cos(x), frac partial f_1 partial y=-sin^2(x)sin(y)$



      $frac partial f_2 partial x=y, frac partial f_2 partial y=x$.



      So the derivative is a $2x2$ matrix $D_f(v)=frac partial f_i partial x_j$. (Use $x_1=x$ and $x_2=y$ for simplicity of writing)






      share|cite|improve this answer























        up vote
        0
        down vote



        accepted







        up vote
        0
        down vote



        accepted






        You can calculate the partial derivatives, and see that they`re continous. This means the function is differentiable .



        $$f_1(x,y)=sin^2(x)cos(y)$$



        $$f_2(x,y)=xy$$



        $frac partial f_1 partial x=cos(y)2sin(x)cos(x), frac partial f_1 partial y=-sin^2(x)sin(y)$



        $frac partial f_2 partial x=y, frac partial f_2 partial y=x$.



        So the derivative is a $2x2$ matrix $D_f(v)=frac partial f_i partial x_j$. (Use $x_1=x$ and $x_2=y$ for simplicity of writing)






        share|cite|improve this answer













        You can calculate the partial derivatives, and see that they`re continous. This means the function is differentiable .



        $$f_1(x,y)=sin^2(x)cos(y)$$



        $$f_2(x,y)=xy$$



        $frac partial f_1 partial x=cos(y)2sin(x)cos(x), frac partial f_1 partial y=-sin^2(x)sin(y)$



        $frac partial f_2 partial x=y, frac partial f_2 partial y=x$.



        So the derivative is a $2x2$ matrix $D_f(v)=frac partial f_i partial x_j$. (Use $x_1=x$ and $x_2=y$ for simplicity of writing)







        share|cite|improve this answer













        share|cite|improve this answer



        share|cite|improve this answer











        answered Jul 30 at 19:54









        Sar

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        3699












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