# Fréchet derivative of the (matrix) exponential function

$$ D\exp(x)u = \int_0^1 e^{sx}ue^{(1-s)x}\,ds. $$ This intriguing formula expresses the derivative of the exponential map on a Banach algebra as an integral. In particular, using “matrix calculus” notation we have the formula $$ d\exp(X)= \int_0^1 e^{sX}(dX)e^{(1-s)X}\,ds $$ when \(X\) is a square matrix. As we’ll see, this is not too hard to prove.

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