## 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...

## 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)\...

## 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...

## 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 ...

## 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 diff...

## Differentiation done correctly: 2. Higher derivatives

Navigation: 1. The derivative | 2. Higher derivatives | 3. Partial derivatives | 4. Inverse and implicit functions | 5. Maxima and minima Last time, we covered ...

## Differentiation done correctly: 1. The derivative

Navigation: 1. The derivative | 2. Higher derivatives | 3. Partial derivatives | 4. Inverse and implicit functions | 5. Maxima and minima In multivariable calcu...