Gradient rules

Multivariate calculus - differentiate expressions with nabla/gradient operator. - 01/2015

Just leaving some notes to differentiate expressions with the $ \nabla $ operator to compute gradients of various functions.

Usual operations

Sum rule, Product rule, sum rule, division rule, scalar rule ($f$ and $g$ both scalar functions $g: \mathbb R^n \rightarrow \mathbb R$, $f: \mathbb R^n \rightarrow \mathbb R$):

$$ \begin{array}{lcl} \nabla [ f + g ] & = &\nabla f + \nabla g \\ \nabla [ f . g ] & = & \nabla f . g + f . \nabla g \\ \nabla \left [ \frac{f}{g} \right ] & = &\frac{\nabla f . g - f . \nabla g}{g^2} \\ \nabla [ \alpha . f ] & = & \alpha . \nabla f \end{array}$$

Gradient of the norm

$$\nabla [ \| \vec {x} \| ] = \frac{ \vec {x}}{ \| \vec {x} \|} $$

$$\nabla [ \| \vec {x} \|^2 ] = 2 . \vec {x} $$

Gradient of a matrix

With $ M $ a $ n \times n $ matrix:

$$\nabla [ M \vec {x} ] = M $$

Likewise for rigid transformations (rotations $ M \in \mathbb R^{3 \times 3}$ and translations $\vec t \in \mathbb R^3$):

$$\nabla [ M \vec {x} + \vec t ] = M $$

Chain rules

With $s: \mathbb R \rightarrow \mathbb R$ univariate and $f: \mathbb R^n \rightarrow \mathbb R$ multivariate real valued the operation boils down to a uniform scale of the gradient:
$$ \nabla \left [ s( f(\vec {x}) ) \right ] = s'( f(\vec {x}) ) \nabla f(\vec {x}) $$

With $m: \mathbb R^n \rightarrow \mathbb R^n$ deformation map and $f: \mathbb R^n \rightarrow \mathbb R$ multivariate scalar function the operation boils down to transform the gradient with a matrix:

$$ \nabla \left [ f( m(\vec {x}) ) \right ] = \mathbf{J}\left [ m(\vec {x}) \right ]^\mathsf{T} \nabla f(m(\vec {x}))$$

Where $ \mathbf{J}\left [ m(\vec {x}) \right ]^\mathsf{T} $ denotes the transpose of the $n \times n $ Jacobian matrix.

Related Jacobian rules

Jacobian of a linear function is the identity matrix $ I $ times the slope $ a $:
$$ J[a \vec {x} + c] = a.I $$

Related chain rule

A univariate differentiation can lead to the use of $ \nabla $: with $f: \mathbb R^n \rightarrow \mathbb R$ multivariable scalar function and $p: \mathbb R \rightarrow \mathbb R^n$ a parametric function:

$$ f'(g(x)) = \nabla f(g(x))^T . \vec {g'(x)} $$

In short, we do the dot product between the gradient of $ f $ and the speed of $ g $

Some links:

Wikipedia says

Multivariate / multivariable chain rule