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I am looking into a problem that involves differentiation of a second order symmetric tensor.

I realize that for a non symmetric tensor $C$ gives $\frac{dC_{ij}}{dC_{kl}}=\delta_{ik}\delta_{jl}$.

Now, given a symmetric tensor I saw on wikipedia that one gets $\frac{1}{2}(\delta_{ik}\delta_{jl}+\delta_{jk}\delta_{il})$

But this does not seem right.

If you take for example a $2\times2$ matrix $A$ full of ones, you get $$\frac{dA_{12}}{dA_{21}}= \frac{1}{2}(\delta_{12}\delta_{21}+\delta_{22}\delta_{11})=\frac{1}{2}$$ when the result should be $1$. Any Ideas?

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