# 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.

# Convex functions, second derivatives and Hessian matrices

In single variable calculus, a twice differentiable function $$f:(a,b)\to\mathbb{R}$$ is convex if and only if $$f^{\prime\prime}(x)\ge 0$$ for all $$x\in(a,b)$$. It is not too hard to extend this result to functions defined on more general spaces:

# Differentiation done correctly: 5. Maxima and minima

Navigation: 1. The derivative | 2. Higher derivatives | 3. Partial derivatives | 4. Inverse and implicit functions | 5. Maxima and minima In this final post, we are going to look at some applications of differentiation to locating maxima and minima of real valued functions. In order to do this, we will be using […]

# Differentiation done correctly: 4. Inverse and implicit functions

Navigation: 1. The derivative | 2. Higher derivatives | 3. Partial derivatives | 4. Inverse and implicit functions | 5. Maxima and minima Now we’re going to prove the inverse function and implicit function theorems for Banach spaces.

# Differentiation done correctly: 3. Partial derivatives

Navigation: 1. The derivative | 2. Higher derivatives | 3. Partial derivatives | 4. Inverse and implicit functions | 5. Maxima and minima While we saw that differentiable maps may be naturally split into component functions when the codomain is a product of Banach spaces, the situation for the domain is more complicated. (This is […]