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  1. #1
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    partial derivative question

    If g(x,y) = ln (ye^xy) g sub x = ? My answers show it is simply y but I do not understand why. I thought since the derv of a ln is 1/(ye^xy) and since y is constant in this case the answer should be 1/e^x?????????
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by Frostking View Post
    If g(x,y) = ln (ye^xy) g sub x = ? My answers show it is simply y but I do not understand why. I thought since the derv of a ln is 1/(ye^xy) and since y is constant in this case the answer should be 1/e^x?????????
    Note that g\!\left(x,y\right)=\ln\!\left(ye^{xy}\right)=\ln\  !\left(y\right) + \ln\!\left(e^{xy}\right)=\ln\!\left(y\right)+xy

    Now, when you partially differentiate with respect to x, y is treated as a constant.

    Thus, \frac{\partial g}{\partial x}=\frac{\partial}{\partial x}\left[\ln y\right]+y\frac{\partial}{\partial x}\left[x\right]=0+y=\color{red}\boxed{y}

    Does this make sense?
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  3. #3
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    differential question answer

    Yes, thank you I did not realize that you should split into two natural logs and I did not know that the ln e^xy is simply xy. Thanks much! Frostking
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