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Math Help - Partial derivatives problem

  1. #1
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    Partial derivatives problem

    1. a) Calculate the partial derivatives for f(x,y)=x^2 siny and prove that

    b)If x(u)=u^2 and y(u)=1/u, calculate df/du using the chain rule.

    c)For g(r)=r^4 where r=sqrt(x^2+y^2+z^2) use the chain rule to calculate dg/dx, d^2g/dx2 and d^2g/dxdy.

    I am having a lot of trouble with this problem.
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    Quote Originally Posted by john1991 View Post
    1. a) Calculate the partial derivatives for f(x,y)=x^2 siny and prove that

    b)If x(u)=u^2 and y(u)=1/u, calculate df/du using the chain rule.

    c)For g(r)=r^4 where r=sqrt(x^2+y^2+z^2) use the chain rule to calculate dg/dx, d^2g/dx2 and d^2g/dxdy.

    I am having a lot of trouble with this problem.
    a) To find the partial derivative of f(x,y) with respect to x, treat y as a constant, and vice versa. Have you tried this?

    b) Maybe someone else can help me out here because I don't see how the chain rule applies. Replace all instances of x with x(u) = u^2 and likewise for y in f(x,y). You are left with a univariate problem of finding the derivative of f(u) with respect to u. (Here we are overloading the variable name f as in: f(u) = f(x(u),y(u)), which may be confusing, but that seems to be the notation of the problem.)

    c) First of all, g(r) simplifies to g(x,y,z)=(x^2+y^2+z^2)^2. For the first partial derivative, you'll solve exactly as you would

    d/dx (x^2 + c)^2

    As in,

    d/dx (x^2 + c)^2 = 2*(x^2 + c) * 2x

    Hopefully it makes more sense now?
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