Maths > Differentiation > 8.0 Matrix Differentiation
Differentiation
1.0 Differentiation
2.0 Some Basic Differentiation formulae
3.0 Properties of Differentiation
4.0 Derivative of Common Functions
4.1 Derivative of Trigonometric functions
4.2 Derivative of Inverse Trigonometric functions
4.3 Derivative of Exponential and Logarithmic Function
4.4 Derivative Of Hyperbolic function
4.5 Questions related to derivation of common functions
5.0 Explicit and Implicit form:
6.0 Parametric Differentiation
7.0 Differentiation of one function w.r.t other
8.0 Matrix Differentiation
9.0 Logarathimic Differentiation
10.0 Differentiation using substitution
8.1 Differentiation of determinant matrix
4.2 Derivative of Inverse Trigonometric functions
4.3 Derivative of Exponential and Logarithmic Function
4.4 Derivative Of Hyperbolic function
4.5 Questions related to derivation of common functions
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, \[y = \left| {\begin{array}{c} {a{'_{11}}}&{a{'_{12}}}&{a{'_{13}}} \\ {{a_{21}}}&{{a_{22}}}&{{a_{23}}} \\ {{a_{31}}}&{{a_{32}}}&{{a_{33}}} \end{array}} \right| + \left| {\begin{array}{c} {{a_{11}}}&{{a_{12}}}&{{a_{13}}} \\ {a{'_{21}}}&{a{'_{22}}}&{a{'_{23}}} \\ {{a_{31}}}&{{a_{32}}}&{{a_{33}}} \end{array}} \right| + \left| {\begin{array}{c} {{a_{11}}}&{{a_{12}}}&{{a_{13}}} \\ {{a_{21}}}&{{a_{22}}}&{{a_{23}}} \\ {a{'_{31}}}&{a{'_{32}}}&{a{'_{33}}} \end{array}} \right|\]
Question 1: Find derivative of \[y = \left| {\begin{array}{c} {{x^2}}&{{x^3}}&x \\ {{e^x}}&{\log x}&{{2^x}} \\ {\cos x}&{\sinh x}&{{{\tan }^{ - 1}}x} \end{array}} \right|\]
Solution: \[y = \left| {\begin{array}{c} {{x^2}}&{{x^3}}&x \\ {{e^x}}&{\log x}&{{2^x}} \\ {\cos x}&{\sinh x}&{{{\tan }^{ - 1}}x} \end{array}} \right|\]
\[\frac{{dy}}{{dx}} = \left| {\begin{array}{c} {\frac{d}{{dx}}{x^2}}&{\frac{d}{{dx}}{x^3}}&{\frac{d}{{dx}}x} \\ {{e^x}}&{\log x}&{{2^x}} \\ {\cos x}&{\sinh x}&{{{\tan }^{ - 1}}x} \end{array}} \right| + \left| {\begin{array}{c} {{x^2}}&{{x^3}}&x \\ {\frac{d}{{dx}}{e^x}}&{\frac{d}{{dx}}\log x}&{\frac{d}{{dx}}{2^x}} \\ {\cos x}&{\sinh x}&{{{\tan }^{ - 1}}x} \end{array}} \right| + \left| {\begin{array}{c} {{x^2}}&{{x^3}}&x \\ {{e^x}}&{\log x}&{{2^x}} \\ {\frac{d}{{dx}}\cos x}&{\frac{d}{{dx}}\sinh x}&{\frac{d}{{dx}}{{\tan }^{ - 1}}x} \end{array}} \right|\]
We know that, $$\frac{d}{dx}x^n = nx^{n-1}$$$$ \frac{d}{{dx}}{e^x} = {e^x} $$$$ \frac{d}{{dx}}\cos x = - \sin x $$$$ \frac{d}{{dx}}{\tan ^{ - 1}}x = \frac{1}{{1 + {x^2}}} $$$$ \frac{d}{{dx}}{a^x} = {a^x}\log a $$$$ \frac{d}{{dx}}{\log _e}x = \frac{1}{{x\ln e}} = \frac{1}{x} $$$$ \frac{d}{{dx}}\sinh x = \cosh x $$
\[\frac{{dy}}{{dx}} = \left| {\begin{array}{c} {2{x^{2 - 1}}}&{3{x^{3 - 1}}}&{1{x^{1 - 1}}} \\ {{e^x}}&{\log x}&{{2^x}} \\ {\cos x}&{\sinh x}&{{{\tan }^{ - 1}}x} \end{array}} \right| + \left| {\begin{array}{c} {{x^2}}&{{x^3}}&x \\ {{e^x}}&{\log x}&{{2^x}\log 2} \\ {\cos x}&{\sinh x}&{{{\tan }^{ - 1}}x} \end{array}} \right| + \left| {\begin{array}{c} {{x^2}}&{{x^3}}&x \\ {{e^x}}&{\log x}&{{2^x}} \\ { - \sin x}&{\cosh x}&{\frac{1}{{1 + {x^2}}}} \end{array}} \right|\]
