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Math Help - Derivative of a general function

  1. #1
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    Derivative of a general function

    Hello,

    I have to find the derivative of a general function f(x,y) where x(s,t) = \frac{s+t}{2} and y(s,t) = \frac{s-t}{2}.

    I have already determined that:
    \frac{\partial f}{\partial s} = \frac{1}{2}(\frac{\partial f}{\partial x} + \frac{\partial f}{\partial y})
    and
    \frac{\partial f}{\partial t} = \frac{1}{2}(\frac{\partial f}{\partial x} - \frac{\partial f}{\partial y})

    So, my question is, how can I use this information to get a value for \frac{\partial d^2f}{\partial s \partial t}, again for a general function?

    Thanks for your help in advance, if any clarification is need please ask.
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  2. #2
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    Quote Originally Posted by jonmondalson View Post
    Hello,

    I have to find the derivative of a general function f(x,y) where x(s,t) = \frac{s+t}{2} and y(s,t) = \frac{s-t}{2}.

    I have already determined that:
    \frac{\partial f}{\partial s} = \frac{1}{2}(\frac{\partial f}{\partial x} + \frac{\partial f}{\partial y})
    and
    \frac{\partial f}{\partial t} = \frac{1}{2}(\frac{\partial f}{\partial x} - \frac{\partial f}{\partial y})

    So, my question is, how can I use this information to get a value for \frac{\partial^2f}{\partial s \partial t}, again for a general function?
    To find the second derivatives with respect to s or t, use the chain rule, just as you did to find the first derivatives. In fact,

    \begin{aligned}\frac{\partial^2f}{\partial s \partial t} = \frac{\partial}{\partial s}\Bigl(\frac{\partial f}{\partial t}\Bigr) &= \frac{\partial}{\partial x}\Bigl(\frac{\partial f}{\partial t}\Bigr)\frac{\partial x}{\partial s} + \frac{\partial}{\partial y}\Bigl(\frac{\partial f}{\partial t}\Bigr)\frac{\partial y}{\partial s} \\ &= \frac{\partial}{\partial x}\Bigl(\tfrac{1}{2}\bigl(\tfrac{\partial f}{\partial x} - \tfrac{\partial f}{\partial y}\bigr)\Bigr)*\tfrac12 + \frac{\partial}{\partial y}\Bigl(\tfrac{1}{2}\bigl(\tfrac{\partial f}{\partial x} - \tfrac{\partial f}{\partial y}\bigr)\Bigr)*\tfrac12 \\ &= \ldots \text{ \footnotesize (I'll leave you to finish it)} .\end{aligned}
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