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I think my calculations of $\nabla g(x) = (x^{T} x)^{m+1}$ are wrong...mostly the last part. Can someone help me?

Given $g(x) = (x^{T} x)^{m+1}$ we say that $f(x) = x^{T}x = \langle x,x \rangle = \sum_{i=1}^{n}x_{i}^{2}$ thus: \begin{equation} \begin{aligned} \nabla f(x) = \begin{pmatrix} {{\partial}\over{\partial x_{1}}}(\sum_{i=1}^{n}x_{i}^{2})\\ {{\partial}\over{\partial x_{2}}}(\sum_{i=1}^{n}x_{i}^{2})\\ \vdots\\ {{\partial}\over{\partial x_{n}}}(\sum_{i=1}^{n}x_{i}^{2}) \end{pmatrix} = \begin{pmatrix} 2x_{1}\\ 2x_{2}\\ \vdots\\ 2x_{n}\\ \end{pmatrix} = 2x \end{aligned} \end{equation} now, we can derive \begin{equation} \nabla g(x) = (m+1)(x^{T}x)^{m}2x \end{equation}

Moleson
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    Its correct, you went from something "like $x^{2m+2}$" to something "like $(2m+2)x^{2m+1}"$, in perfect analogy with the dimension 1 case – Calvin Khor Oct 09 '18 at 18:00
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    You have the correct result. Use the chain rule with $h:y\mapsto y^{m+1}$, $f:x\mapsto x^Tx$ and $g=h\circ f$. – amd Oct 09 '18 at 18:13

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