Multivariate derivatives cheatsheet

Calculus

We provide an easy to use reference for oft-used multivariate derivatives.

Prerequisites

To understand and use these derivatives, we recommend familiarity with the concepts in

Follow the above links to first get acquainted with the corresponding concepts.

Some oft-used derivatives

Input Gradient
\( \partial \mA \) 0
\( \partial (\alpha \mX) \) \( \alpha \partial \mX \)
\( \partial (\mX + \mY) \) \( \partial \mX + \partial \mY \)
\( \partial (\mX\mY) \) \( (\partial \mX)\mY + \mX(\partial \mY) \)
\( \partial \inv{\mX} \) \( -\inv{\mX} \left( \partial \mX\right) \inv{\mX} \)
\( \partial \mX^T \) \( \left(\partial \mX\right)^T \)
\( \partial \vx^T \va \) \( \va \)
\( \partial \left(\va^T \mX \vb\right) \) \( \va^T \vb \)
\( \partial \left(\va^T \mX^T \vb\right) \) \( \vb \va^T \)
\( \partial \left(\norm{\vx - \va}{2}\right) \) \( \frac{\vx - \va}{\norm{\vx - \va}{2}} \)
\( \partial \left(\norm{\mX}{2}\right) \) \( 2\mX \)

Where to next?

This was a quick reference for multivariate derivatives. Explore our other comprehensive articles on topics in calculus.

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