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Math Help - Partial derivative, multivariable

  1. #1
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    Partial derivative, multivariable

    f(x,y)

    For all s,t :
    f(s,8s)=6s+cos(6s)
    f(t,-7t)=1+6t+5t^2

    Determin \frac{\partial f}{\partial x}(0,0) and \frac{\partial f}{\partial y}(0,0)

    Im new to multi variables so i donīt know how to do this.

    I thought if i set z=f(x(s,t),y(s,t))

    i get:

    \frac{\partial z}{\partial s }= \frac{\partial z}{\partial x}\frac{\partial x}{\partial s}+\frac{\partial z}{\partial y}\frac{\partial y}{\partial s}

    \frac{\partial z}{\partial t }= \frac{\partial z}{\partial x}\frac{\partial x}{\partial t}+\frac{\partial z}{\partial y}\frac{\partial y}{\partial t}

    from here i donīt know what to do.

    Thanks!
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  2. #2
    Behold, the power of SARDINES!
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    Quote Originally Posted by mechaniac View Post
    f(x,y)

    For all s,t :
    f(s,8s)=6s+cos(6s)
    f(t,-7t)=1+6t+5t^2

    Determin \frac{\partial f}{\partial x}(0,0) and \frac{\partial f}{\partial y}(0,0)

    Im new to multi variables so i donīt know how to do this.

    I thought if i set z=f(x(s,t),y(s,t))

    i get:

    \frac{\partial z}{\partial s }= \frac{\partial z}{\partial x}\frac{\partial x}{\partial s}+\frac{\partial z}{\partial y}\frac{\partial y}{\partial s}

    \frac{\partial z}{\partial t }= \frac{\partial z}{\partial x}\frac{\partial x}{\partial t}+\frac{\partial z}{\partial y}\frac{\partial y}{\partial t}

    from here i donīt know what to do.

    Thanks!
    You have a good start. Using the chain rule that you have above gives

    \frac{\partial f}{\partial s}=6-6\sin(6s)=\frac{\partial f}{\partial x}\cdot 1+\frac{\partial f}{\partial y}\cdot 8

    Using the subscript notation for partial derivatives gives
    f_x(s,8s)+8f_y(s,8s)=6-6\sin(6s)

    \frac{\partial f}{\partial t}=6+10t=\frac{\partial f}{\partial x}\cdot 1+\frac{\partial f}{\partial y}\cdot (-7)

    f_x(t,-7t)-7f_y(t,-7t)=6+10t

    Now if
    t=s=0

    You will have a system of equations to find the partial derivatives you want!
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  3. #3
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    Thanks! i just got confused by all the variables
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