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Math Help - Differentiability of Gradient

  1. #1
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    Differentiability of Gradient

    I'm having a bit of difficulty with this problem.

    Show that gradient(f^r) = r*f^(r-1)*gradient(f)

    Not even sure where to begin with this, thanks!
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  2. #2
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    Quote Originally Posted by pakman View Post
    I'm having a bit of difficulty with this problem.

    Show that gradient(f^r) = r*f^(r-1)*gradient(f)

    Not even sure where to begin with this, thanks!
    Let us assume f(x_1,...,x_n) is a differenciable function.

    Thus,
    \nabla f=\left< \frac{\partial f}{\partial x_1},..., \frac{\partial f}{\partial x_n} \right>
    Thus,
    \nabla f^r = \left< \frac{\partial f^r}{\partial x_1},...,\frac{\partial f^r}{\partial x_n} \right>.
    But, by the chain rule,
    \frac{\partial f^r}{\partial x_i}=rf^{r-1}\frac{\partial f}{\partial x_i} for i=1,2,...,n.
    Thus,
    \nabla f^r = \left< rf^{r-1} \frac{\partial f}{\partial x_1},...,rf^{r-1}\frac{\partial f}{\partial x_n} \right>=rf^{r-1}\left< \frac{\partial f}{\partial x_1},...,\frac{\partial f}{\partial x_n} \right>=rf^{r-1}\nabla f
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