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I need to calculate the Jacobian $\frac{df}{dx}$ of $f=A^{-1}b$ where $A$ and $b$ are a function of $x$, the variable towards to differentiate.

I thought

$$\frac{df}{dx} = \frac{dA^{-1}}{dx} b + A^{-1}\frac{db}{dx}$$

by the product rule, and since $A^{-1}A=I$,

$$\frac{dA^{-1}}{dx} = A^{-1} \frac{dA}{dx} A^{-1}.$$

Now the last thing i thought is

$$\frac{dA}{dx} = \frac{dA}{dx_1} + \frac{dA}{dx_2} + \frac{dA}{dx_3} + \cdots $$

The last step is to calculate the Jacobian of a matrix. However, if I try this for a simple example, I get a wrong answer. Can anyone see where I make the mistake? How can I calculate the Jacobian of $f=A^{-1}b$ correct if I cannot analytically invert $A$ (I can only do that numerically)?

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    The trouble is that $dA^{-1}/dx$ needs to be a kind of rank-3 object that when multiplied by the vector $b$ gives a matrix. You would be better off doing this in some form of [index notation](https://en.wikipedia.org/wiki/Einstein_notation).2012-07-27

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