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Thread: Need help to find the first order partial derivatives

  1. #1
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    Need help to find the first order partial derivatives

    This is the given problem I have:
    Use the Chain Rule (or the Invariance Property of differentials) to find the first order partial derivatives (i.e. ∂g/∂s and ∂g/∂t)
    of g(s,t) := f(s,u(s,t),v(s,t)), where
    u(s,t) := st,
    v(s,t) := s + t.
    The answer should be expressed in terms of s and t only.

    I'm completely stuck on this problem, due to the Equation itself. Dont really know how to do it, anyone that can help me?
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  2. #2
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    Re: Need help to find the first order partial derivatives

    Well, you use the phrase "Chain Rule" so apparently you know what that is! Here, g, a function of s and t, is equal to f, a function of the three variables, s, u, and v, with u and v themselves functions of s and t.

    By the "Chain Rule", \frac{\partial g}{\partial s}= \frac{\partial f}{\partial s}+ \frac{\partial f}{\partial u}\frac{\partial u}{\partial s}+ \frac{\partial f}{\partial v}\frac{\partial v}{\partial s}
    (Unfortunate notation but I don't know any better! The first \frac{\partial f}{\partial s} is just the derivative of f with respect to s, ignoring u and v and their dependence on s)

    This is \frac{\partial g}{\partial s}= \frac{\partial f}{\partial s}+ \frac{\partial f}{\partial u}(t)+ \frac{\partial f}{\partial v}(1).
    Do the partial derivative of g with respect to t in the same way.

    You can't do more than this without knowing what "f" is.
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