Derivatives in vector form (gradients)

Summary

• \(f\) is a real valued function of \(m\) variables \(x={\left( x_1, \ldots ,x_m \right)}'\) , \(f:R^m→R\) .
• The derivative of \(f\) with respect to the vector \(x\) , denoted by \( \frac{∂f}{∂x}\) , is defined as the \(m×1\) vector

\[ \frac{∂f}{∂x}=\begin{bmatrix} \frac{∂f}{∂x_1} \\ ⋮ \\ \frac{∂f}{∂x_m}\end{bmatrix}\]

• This vector is called the gradient of \(f\) .
• Example: \(f\left( x_1,x_2 \right)=2x_1+3x_2\) then

\[ \frac{∂f}{∂x}=\begin{bmatrix}2 \\ 3\end{bmatrix}\]

• Result: derivative of a linear function. If \(f\left( x \right)=a'x\) then

\[ \frac{∂f}{∂x}=a\]

• Result: derivative of a quadratic form. If \(f\left( x \right)=x'Ax\) and \(A\) is symmetric then

\[ \frac{∂f}{∂x}=2Ax\]