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Math Help - Derivative of arctan in a partial derivative

  1. #1
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    Derivative of arctan in a partial derivative

    Im working on the derivative f'y(x,y) of the function f(x,y)=arctan(x^2)+ y^3

    Because it's a derivative with respect to the variable y, this means that the variable x is held constant, does this mean that the entire arctan(x^2) should be regarded as just a number so the partial derivative equals to 3y^2 in the end??
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  2. #2
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    Quote Originally Posted by tinyone View Post
    Im working on the derivative f'y(x,y) of the function f(x,y)=arctan(x^2)+ y^3
    Because it's a derivative with respect to the variable y, this means that the variable x is held constant, does this mean that the entire arctan(x^2) should be regarded as just a number so the partial derivative equals to 3y^2 in the end??
    You are correct.
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  3. #3
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    Cheers,

    How about this one:

    Derivative f'z(x,y,z) = 2(z+3)^4 + ln(xyz)

    So keeping y and x constant would give me: f'z= 8(z+3)^3 + (1/xyz) * xy
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  4. #4
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    Quote Originally Posted by tinyone View Post
    Cheers,

    How about this one:

    Derivative f'z(x,y,z) = 2(z+3)^4 + ln(xyz)

    So keeping y and x constant would give me: f'z= 8(z+3)^3 + (1/xyz) * xy
    Of course, you would want to cancel the "xy" terms in numerator and denominator of that last fraction to get
    f_z= 8(z+ 3)^3+ \frac{1}{z}
    You could see that more easily by writing f(x, y, z()= 2(z+ 3)^4+ ln(xyz)= 2(z+ 3)^4+ ln(x)+ ln(y)+ ln(z).

    Oh, and don't write "f'z" for both the function and its partial derivative!

    In fact, it is not a good idea to use the ' with partial derivatives. You problem is to find the derivative, with respect to z,
    of f(x,y,z)= 2(z+ 3)^4+ ln(xyz) and the correct answer, in simplest form, is f_z= 8(z+ 3)^3+ 1/z.
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