Derivative of A Quadratic Form with a Change of Basis

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The derivative of a quadratic form is given by:

$\Delta_{X} X^{T}AX = 2AX$

where A is symmetric.

What is the derivative of the following:

$\Delta_{X} (DX)^{T}A(DX)$,

where $A$ is positive and symmetric and $D$ is not necessarily symmetric?

I would like to rearrange terms such that the $D$ is on the inside of the operator, however, I am unsure of whether I can do this.

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Since $D^TAD$ is symmetric, it seems you can replace $A$ in your original result by $D^TAD$.