Vector derivative of expression involving square root

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How do I differentiate $\sqrt{\mathbf{x}^TM\mathbf{x}}$ with respect to vector $\mathbf{x}$ when $M$ is a symmetric matrix?

I guess that some generalization of the chain rule could be used, but I do not know how. I have managed to show that the derivative of $\mathbf{x}^TM\mathbf{x}$ with respect to $\mathbf{x}$ is $(M+M^T)\mathbf{x}$ but I do not know how to continue.