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Math Help - Formula for gradient of composite function

  1. #1
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    Formula for gradient of composite function

    Suppose that f: \mathbb {R}^n \rightarrow \mathbb {R} and g: \mathbb {R} \rightarrow \mathbb {R}

    Find a formula for  \nabla (g \circ f)(x)

    I have  \nabla (g \circ f) = ( \frac { \partial (g \circ f) }{ \partial x_1 }(x),..., \frac { \partial (g \circ f) }{ \partial x_n }(x) )

    =  ( \frac { \partial g (f(x)) }{ \partial x_1} \frac { \partial f}{ \partial x_1 }(x),...,\frac { \partial g (f(x)) }{ \partial x_n} \frac { \partial f}{ \partial x_n }(x))
    = \nabla g(f(x)) \nabla f(x) ,

    is this right? Thanks.
    Last edited by tttcomrader; April 22nd 2009 at 08:35 PM.
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  2. #2
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    Quote Originally Posted by tttcomrader View Post
    Find a formula for  \nabla (g \circ f)(x)

    I have  \nabla (g \circ f) = ( \frac { \partial (g \circ f) }{ \partial x_1 }(x),..., \frac { \partial (g \circ f) }{ \partial x_n }(x) )

    =  ( \frac { \partial g (f(x)) }{ \partial x_1} \frac { \partial f}{ \partial x_1 }(x),...,\frac { \partial g (f(x)) }{ \partial x_n} \frac { \partial f}{ \partial x_n }(x))
    = \nabla g(f(x)) \nabla f(x) ,

    is this right? Thanks.
    i don't think so! since g \circ f is a map from a subset of \mathbb{R}^n to \mathbb{R}, we need f to be a map from a subset of \mathbb{R}^n to \mathbb{R} and g a map from the image of f, which is a subset of \mathbb{R}, to \mathbb{R}.
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