Maths for Design Optimisation: Computing Derivatives

Master Derivatives Computation Techniques for Gradient-Based Methods in Multidisciplinary Design Optimisation

Gradients and curvature sit at the heart of many of the most powerful optimisation algorithms used in engineering. This course focuses on helping you understand what derivatives really mean in an optimisation context — and how they are computed in practice.

In this hands-on course, you’ll develop an intuitive and practical understanding of derivatives for design optimisation, moving beyond textbook calculus to see how gradients and curvature shape optimisation behaviour. You’ll explore how derivatives describe local sensitivity, search directions, and landscape geometry — and why accurate derivative information is so important for efficient optimisation.

Starting from first principles, we’ll build geometric intuition around gradients, directional derivatives, and curvature, gradually extending this understanding to second-order information such as the Hessian and principal curvatures. Rather than treating these concepts abstractly, you’ll learn how to interpret them directly on optimisation landscapes and understand their role in guiding algorithms toward optimal solutions.

The course then turns to the practical question of how derivatives are actually computed in real optimisation workflows. You’ll study and compare different differentiation strategies, including black-box approaches such as finite differencesand the complex-step method, as well as white-box approaches like symbolic or algorithmic differentiation. You’ll also explore grey-box methods, including implicit analytic techniques, which are widely used in Multidisciplinary Design Optimisation.

As with the rest of the series, the emphasis is on intuition and application rather than abstract derivations. You’ll work through hands-on coding exercises in Python, computing and visualising derivatives, comparing accuracy and efficiency across methods, and revisiting realistic engineering examples — including a return to Kepler’s equation from earlier courses.

By the end of this course, you’ll:

• Understand gradients and curvature from an optimisation perspective

• Be able to interpret directional derivatives, Hessians and principal curvatures geometrically

• Know the strengths and limitations of different derivative computation methods

• Compare black-box, white-box, and grey-box differentiation approaches

• Gain hands-on experience computing derivatives using Scipy, Sympy, Jax and OpenMDAO

• Be prepared to use derivative information effectively in gradient-based optimisation algorithms

This course is designed for engineers, students, and technical professionals who want to understand where optimisation gradients come from and how to compute them reliably — especially if you plan to use advanced, gradient-based optimisation methods in practice.

A basic familiarity with mathematical optimisation is recommended, as this course builds directly on earlier modules in the Maths for Design Optimisation series.

If you’re ready to demystify derivatives and build strong intuition for gradient-based optimisation, this course sets the foundation for the advanced algorithms that follow.