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Thread: 3 questions on partial derivatives

  1. #1
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    3 questions on partial derivatives

    Find the first partial derivatives of the function:

    1) $\displaystyle f(x,y) = x^y$

    2) $\displaystyle f(x,y) = \int_{y}^{x} cos(t^2) dt$

    3) $\displaystyle u = x^{\frac{y}{z}}$

    Solutions

    1) $\displaystyle \frac{\partial f}{\partial x} = yx^{y-1}$ , $\displaystyle \frac{\partial f}{\partial y} = ln(x)$

    2) I have no idea

    3) $\displaystyle \frac{\partial u}{\partial x} = \frac{y}{z}x^{\frac{y}{z}-1}$ , $\displaystyle \frac{\partial u}{\partial y} = \frac{ln(x)}{z} = x^{\frac{1}{z}}$ , $\displaystyle \frac{\partial u}{\partial z} = \frac{yln(x)}{z^2} = x^{\frac{y}{z^2}}$

    are these correct, and any idea on how to solve 2 would be greatly appreciated.
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  2. #2
    Behold, the power of SARDINES!
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    $\displaystyle
    f(x,y) = \int_{y}^{x} cos(t^2) dt
    $

    by FTC we get...

    $\displaystyle \frac{\partial{f}}{\partial{x}}=\cos(x^2)$

    for y we need to rewrite the integtal so that the FTC applies

    $\displaystyle
    f(x,y) = \int_{y}^{x} cos(t^2) dt=-\int_{x}^{y} \cos(t^2)dt
    $

    Now

    $\displaystyle \frac{\partial{f}}{\partial{y}}=-cos(y^2)$
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  3. #3
    Senior Member Peritus's Avatar
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    1. $\displaystyle \frac{{\partial f}}
    {{\partial y}} = x^y \ln x
    $

    3. $\displaystyle \frac{{\partial u}}
    {{\partial y}} = \frac{1}
    {z}x^{\frac{y}
    {z}} \ln x$

    $\displaystyle
    \frac{{\partial u}}
    {{\partial z}} = - \frac{y}
    {{z^2 }}x^{\frac{y}
    {z}} \ln x
    $
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  4. #4
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
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    1. has an error

    $\displaystyle f(x,y)=x^y \iff ln(f(x,y))=y \ln(x)$ taking the derivative we get

    $\displaystyle \frac{1}{f(x,y)}\frac{\partial{f}}{\partial{y}}=\l n(x)$

    so

    $\displaystyle \frac{\partial{f}}{\partial{y}}=f(x,y) \cdot \ln(x) =x^y \ln(x)$
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