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Mastering Optimization with Python

Second-Order Derivatives and the Hessian

Explore second-order derivatives as derivatives of derivatives and how they apply to functions with multiple variables. Learn to compute partial derivatives and construct the Hessian matrix with Python code. This lesson guides you through calculating second-order derivatives, including cross-derivatives, and illustrates the importance of variable order in derivatives for solving optimization problems.

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As derivatives are functions, they have their own derivatives. The derivative of a derivative is called a second-order derivative. In the following code, we calculate the second-order derivative of f=x2f = x^2.

Python 3.10.4
from sympy import symbols, diff
x = symbols('x')
f = x**2
df = diff(f, x) # first-order derivative
ddf =  diff(df, x) # second-order derivative
print("First-order:", df)
print("Second-order:", ddf)

Second-order derivatives in multiple dimensions

In cases with more than one variable, we can calculate the second-order derivative with respect to the same first variable or with respect to another variable. For example, we denote as ∂2f∂x2\frac{\partial^2 f}{\partial x^2} the operation of taking the derivative of ff with respect to xx and then taking again the derivative with respect to xx. But we could take the second derivative with respect to another variable. We denote as ∂2f∂x∂y\frac{\partial^2 f}{\partial x\partial y} when we first take the derivative with respect to xx and then with respect to yy. See an example with ...