We provide an easy to use reference for oft-used multivariate derivatives.
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.
Input | Gradient |
---|---|
\( \partial \mA \) | 0 |
\( \partial (\alpha \mX) \) | \( \alpha \partial \mX \) |
\( \partial (\mX + \mY) \) | \( \partial \mX + \partial \mY \) |
\( \partial (\mX\mY) \) | \( (\partial \mX)\mY + \mX(\partial \mY) \) |
\( \partial \inv{\mX} \) | \( -\inv{\mX} \left( \partial \mX\right) \inv{\mX} \) |
\( \partial \mX^T \) | \( \left(\partial \mX\right)^T \) |
\( \partial \vx^T \va \) | \( \va \) |
\( \partial \left(\va^T \mX \vb\right) \) | \( \va^T \vb \) |
\( \partial \left(\va^T \mX^T \vb\right) \) | \( \vb \va^T \) |
\( \partial \left(\norm{\vx - \va}{2}\right) \) | \( \frac{\vx - \va}{\norm{\vx - \va}{2}} \) |
\( \partial \left(\norm{\mX}{2}\right) \) | \( 2\mX \) |
This was a quick reference for multivariate derivatives. Explore our other comprehensive articles on topics in calculus.
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