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Math Help - chain rule with partial derivatives

  1. #1
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    chain rule with partial derivatives

    Would someone please take the partial derivatives for x and y for the following:

    f(x,y) = [3x2y][(x+7)^(x/y)]

    Thanks.
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    Re: chain rule with partial derivatives

    Quote Originally Posted by isaac View Post
    Would someone please take the partial derivatives for x and y for the following:

    f(x,y) = [3x2y][(x+7)^(x/y)]

    Thanks.
    Hint: When taking a partial derivative with respect to one variable, all other variables are treated as constants.
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  3. #3
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    Re: chain rule with partial derivatives

    i know about holding the other variables constant. i am getting tripped up on the x/y exponent as well as how to implement the chain rule. could you possibly work out the partial derivatives for x and y for me?
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    Re: chain rule with partial derivatives

    Yes I could, but I won't, as it goes against MHF policy. For another hint, when dealing with variables in your exponents, you should take the logarithm of both sides and simplify using logarithm rules, before differentiating implicitly.
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    Re: chain rule with partial derivatives

    You can also use the fact that x^y = e^(y*ln(x))
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    Re: chain rule with partial derivatives

    Or, use "logarithmic differentiation". With f(x,y) = [3x2y][(x+7)^(x/y)]= 6xy(x+7)^(x/y) (unless that "[3x2y]" means something completely different, like "[3x^3y]") we have
    ln(f)= ln((6xy)(x+7)^(x/y))= ln(6)+ ln(x)+ ln(y)+ (x/y)ln(x+7)

    Now, differentiating on both sides with respect to x, say,
    (1/f) f_x= 1/x + (1/y)ln(x+7)+ x/(y(x+7) and we can solve for f_x by multiplying both sides by the original function, f.

    The derivative with respect to y can be done similarly.
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