Chain Rule
Overview
Single Variable
{% [f(g(x))]' = f'(g(x)) \times g'(x) %}
Multi Variable Chain Rule
{%
\frac{\partial{\vec{f}}}{\partial{\vec{x}}} =\begin{bmatrix}
\frac{\partial{f_1}}{\partial{x_1}} & \frac{\partial{f_1}}{\partial{x_2}} & ... & \frac{\partial{f_1}}{\partial{x_n}} \\
\frac{\partial{f_2}}{\partial{x_1}} & \frac{\partial{f_2}}{\partial{x_2}} & ... & \frac{\partial{f_2}}{\partial{x_n}} \\
... \\
\frac{\partial{f_m}}{\partial{x_1}} & \frac{\partial{f_m}}{\partial{x_2}} & ... & \frac{\partial{f_m}}{\partial{x_n}} \\
\end{bmatrix}
%}
{% \frac{\partial{\vec{f}}}{\partial{\vec{x}}} = \frac{\partial{\vec{f}}}{\partial{\vec{g}}} \times \frac{\partial{\vec{g}}}{\partial{\vec{x}}} %}
Examples
