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Math Help - 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) f(x,y) = x^y

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

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

    Solutions

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

    2) I have no idea

    3) \frac{\partial u}{\partial x} = \frac{y}{z}x^{\frac{y}{z}-1} , \frac{\partial u}{\partial y} = \frac{ln(x)}{z} = x^{\frac{1}{z}} , \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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    <br />
f(x,y) = \int_{y}^{x} cos(t^2) dt<br />

    by FTC we get...

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

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

    <br />
f(x,y) = \int_{y}^{x} cos(t^2) dt=-\int_{x}^{y} \cos(t^2)dt<br />

    Now

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

    3. \frac{{\partial u}}<br />
{{\partial y}} = \frac{1}<br />
{z}x^{\frac{y}<br />
{z}} \ln x

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

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

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

    so

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