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Thread: Partial Derivatives Of Composite Function

  1. #1
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    Partial Derivatives Of Composite Function

    Question: Let $\displaystyle x(t_1, t_2) = t_1 e^{t_2}, y(t_1, t_2) = t_1^2 + sin(t_1 t_2) $. Let $\displaystyle f(x,y) $ be a differentiable function $\displaystyle f: \mathbb{R}^2 \rightarrow \mathbb{R} $. Let $\displaystyle g(t_1, t_2) = f( x(t_1, t_2), y(t_1, t_2) ) $. Express $\displaystyle \frac{ \partial g}{\partial t_1} (1,0) $ in terms of partial derivatives of $\displaystyle f $.

    My Solution: Let $\displaystyle h: \mathbb{R}^2 \rightarrow \mathbb{R}^2 $ be defined as $\displaystyle h(t_1, t_2) = (x(t_1,t_2), y(t_1, t_2)) = (t_1e^{t_2}, t_1^2 + sin(t_1 t_2)) $. Then $\displaystyle g(t_1, t_2) = f(x(t_1, t_2), y(t_1, t_2)) = f(h(t_1, t_2)) $. Note that $\displaystyle f $ is differentiable and so is $\displaystyle h $, because its component functions are differentiable. Thus their composition, $\displaystyle f(h(t_1,t_2)) = g(t_1, t_2) $ is differentiable. Note that $\displaystyle dfh(1,0) = df(h(1,0)) dh(1,0) $. Observe that $\displaystyle dfh(1,0) = \begin{pmatrix} \frac{\partial fh}{\partial t_1}(1, 0) & \frac{\partial fh}{\partial t_2} (1,0) \end{pmatrix} $.


    We also know that $\displaystyle df(h(1,0)) = \begin{pmatrix} \frac{\partial f}{\partial x} h(1,0) & \frac{\partial f}{\partial y} h(1,0) \end{pmatrix} $ and $\displaystyle dh(1,0) = \begin{pmatrix} 1 & 2\\ 1 & 1 \end{pmatrix} $.

    Thus $\displaystyle \begin{pmatrix} \frac{\partial fh}{\partial t_1}(1,0) & \frac{\partial fh}{\partial t_2} (1,0) \end{pmatrix} = \begin{pmatrix} \frac{\partial f}{\partial x} h(1,0) & \frac{\partial f}{\partial y} h(1,0) \end{pmatrix} \begin{pmatrix} 1 & 2\\ 1 & 1 \end{pmatrix} $

    $\displaystyle = \begin{pmatrix} \frac{\partial f}{\partial x}(1,0) + \frac{\partial f}{\partial y}(1,0) & 2 \frac{\partial f}{\partial x}(1,0) + \frac{\partial f}{\partial y}(1,0) \end{pmatrix} $

    Thus $\displaystyle \frac{\partial g}{\partial t_1}(1,0) = \frac{\partial fh}{\partial t_1}(1,0) = \frac{\partial f}{\partial x} (1,0) + \frac{\partial f}{\partial y} (1,0) $.

    Is this correct? Also, is there a faster way to do this? Also, one last question...are the partial derivatives of f taken with respect to x and y or with respect to t_1 and t_2 ? Or are they just dummy variables that I can interchange?
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  2. #2
    Member HappyJoe's Avatar
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    It looks almost perfect. It sounds like you know what's going on. I didn't check you calculation of $\displaystyle dh(1,0)$, but this is not the hardest part anyway.

    Quote Originally Posted by JG89 View Post
    Thus $\displaystyle \begin{pmatrix} \frac{\partial fh}{\partial t_1}(1,0) & \frac{\partial fh}{\partial t_2} (1,0) \end{pmatrix} = \begin{pmatrix} \frac{\partial f}{\partial x} h(1,0) & \frac{\partial f}{\partial y} h(1,0) \end{pmatrix} \begin{pmatrix} 1 & 2\\ 1 & 1 \end{pmatrix} $

    $\displaystyle = \begin{pmatrix} \frac{\partial f}{\partial x}(1,0) + \frac{\partial f}{\partial y}(1,0) & 2 \frac{\partial f}{\partial x}(1,0) + \frac{\partial f}{\partial y}(1,0) \end{pmatrix} $

    Thus $\displaystyle \frac{\partial g}{\partial t_1}(1,0) = \frac{\partial fh}{\partial t_1}(1,0) = \frac{\partial f}{\partial x} (1,0) + \frac{\partial f}{\partial y} (1,0) $.

    Is this correct? Also, is there a faster way to do this? Also, one last question...are the partial derivatives of f taken with respect to x and y or with respect to t_1 and t_2 ? Or are they just dummy variables that I can interchange?
    Note that in the expression

    $\displaystyle \begin{pmatrix} \frac{\partial f}{\partial x}(1,0) + \frac{\partial f}{\partial y}(1,0) & 2 \frac{\partial f}{\partial x}(1,0) + \frac{\partial f}{\partial y}(1,0) \end{pmatrix} $

    you forget to apply $\displaystyle h$ to (1,0) (so that instead of e.g. $\displaystyle \frac{\partial f}{\partial x}(1,0)$, you want $\displaystyle \frac{\partial f}{\partial x}h(1,0) $).

    As for a faster way of solving this, I guess you could save a wee bit of time, but it's not a method I would utilize in a small problem like this. Specifically, you are being asked only for $\displaystyle \frac{\partial g}{\partial t_1}$. For calculating this, you do not need the second column of the matrix for $\displaystyle dh(1,0)$, so you could save yourself some work by calculating only the first column.

    In this problem, the partial derivatives of $\displaystyle f$ are being taken with respect to the variables $\displaystyle x$ and $\displaystyle y$. Since $\displaystyle x$ and $\displaystyle y$ are, in turn, functions of the variables $\displaystyle t_1,t_2$, you could also consider $\displaystyle f$ as a function of $\displaystyle t_1$ and $\displaystyle t_2$. But the partial derivatives of $\displaystyle f$ with respect to $\displaystyle t_1$ and $\displaystyle t_2$ will be completely different from the partial derivatives of $\displaystyle f$ with respect to $\displaystyle x$ and $\displaystyle y$.
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