For a scalar function of several variables, second partial derivatives describe how the first derivatives themselves change.

They are collected in the Hessian matrix

$$H_f(x)=\left[\frac{\partial^2 f}{\partial x_i\partial x_j}\right].$$

For $f(x,y)$,

$$H_f=\begin{bmatrix}f_{xx}&f_{xy}\f_{yx}&f_{yy}\end{bmatrix}.$$

Near a point $x$, a twice-differentiable function has the second-order approximation

$$f(x+d)\approx f(x)+\nabla f(x)^Td+\frac12 d^T H_f(x)d.$$

The gradient describes local slope; the Hessian describes local curvature.

When the mixed partial derivatives are continuous, $f_{ij}=f_{ji}$, so the Hessian is symmetric.