Differentiation
8.0 Matrix Differentiation
8.0 Matrix Differentiation
If, \[y = \left| {\begin{array}{c} {{a_{11}}}&{{a_{12}}}&{{a_{13}}} \\ {{a_{21}}}&{{a_{22}}}&{{a_{23}}} \\ {{a_{31}}}&{{a_{32}}}&{{a_{33}}} \end{array}} \right|\]then, \[y = \left| {\begin{array}{c} {{a'_{11}}}&{{a'_{12}}}&{{a'_{13}}} \\ {{a'_{21}}}&{{a'_{22}}}&{{a'_{23}}} \\ {{a'_{31}}}&{{a'_{32}}}&{{a'_{33}}} \end{array}} \right|\]
Question 1: Find the derivative of \[y = \left[ {\begin{array}{c} {\cos x}&{{e^x}} \\ {2{x^2}}&{\tan x} \end{array}} \right]\]
Solution: \[y = \left[ {\begin{array}{c} {\cos x}&{{e^x}} \\ {2{x^2}}&{\tan x} \end{array}} \right]\]
\[\frac{{dy}}{{dx}} = \left[ {\begin{array}{c} {\frac{d}{{dx}}\cos x}&{\frac{d}{{dx}}{e^x}} \\ {\frac{d}{{dx}}2{x^2}}&{\frac{d}{{dx}}\tan x} \end{array}} \right]\]
We know that, $$\frac{d}{dx}x^n = nx^{n-1}$$$$ \frac{d}{{dx}}{e^x} = {e^x} $$$$ \frac{d}{{dx}}\cos x = - \sin x $$ $$ \frac{d}{{dx}}\tan x = {\sec ^2}x $$
\[\frac{{dy}}{{dx}} = \left[ {\begin{array}{c} { - \sin x}&{{e^x}} \\ {2\left( 2 \right){x^{2 - 1}}}&{{{\sec }^2}x} \end{array}} \right]\]
\[\frac{{dy}}{{dx}} = \left[ {\begin{array}{c} { - \sin x}&{{e^x}} \\ {4{x^1}}&{{{\sec }^2}x} \end{array}} \right]\]
