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Thread: Partial derivatives problem

  1. #1
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    Partial derivatives problem

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

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

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

    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 $\displaystyle f(x,y)=x^2 siny$ and prove that

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

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

    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 $\displaystyle g(x,y,z)=(x^2+y^2+z^2)^2$. 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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