[tex] \texttt{6. Find \( \tt\dfrac{d y}{d x} \), if : }\\ \\ \texttt{(iii) \( \tt x^{y}+y^{x}=\log a \)}[/tex]
[tex] \textsf{Let \( \bf u=x^{y} \) and \( \bf v=y^{x} \Rightarrow u+v=\log a \)}[/tex]
[tex] \pmb{ \[ \begin{array}{ll} \\ \\ \Rightarrow \quad &\displaystyle\sf \frac{d u}{d x}+\frac{d v}{d x}=0 \qquad\qquad\qquad\cdots\cdots (i)\\ \\ \\ &\displaystyle\sf \log u =y \log x \\ \\ \\ \Rightarrow \quad &\displaystyle\sf \frac{1}{ u } \frac{ du }{ d x}=\frac{y}{x}+\log x \frac{ dy }{ d x} \\ \\ \\ \Rightarrow \quad &\displaystyle\sf \frac{ du }{ d x}=x^{y}\left[\frac{y}{x}+\log x \frac{ dy }{ d x}\right] \\ \\ \\ & \displaystyle\sf \log v =x \log y \\ \\ \\ \Rightarrow \quad& \displaystyle\sf \frac{1}{ v } \frac{ dv }{ dx }=\frac{x}{y} \frac{d y}{ d x}+\log y \\ \\ \\ \Rightarrow \quad &\displaystyle\sf \frac{d v}{ d x}=y^{z}\left[\frac{x}{y} \frac{d y}{d x}+\log y\right] \end{array} \] } [/tex]
[tex]\pmb{ \begin{array}{ll} \\ \\ \Rightarrow & \displaystyle \sf y x^{y-1}+x^{y} \log x \frac{ d y}{ d x}+x y^{x-1} \frac{ d y}{ d x}+y^{x} \log y=0 \qquad\texttt{[From (i)]}\\ \\ \\ \blue\Rightarrow & \red{ \displaystyle\sf \frac{d y}{ d x}=-\left[\frac{y \cdot x^{y-1}+y^{x} \cdot \log y}{x y^{x-1}+x^{y} \cdot \log x}\right]}\end{array} }[/tex]
[tex]\\\\\rule{170pt}{0.1pt} \\ \\\small \fcolorbox{magenta}{lavender}{ \color{darkcyan} \textrm{Answerd by ★彡[@ᎥᏖᏕᎩᎧᏬᏒᎮᏗᎶᏝᏗ02]彡★}}[/tex]
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QUESTION :-
[tex] \texttt{6. Find \( \tt\dfrac{d y}{d x} \), if : }\\ \\ \texttt{(iii) \( \tt x^{y}+y^{x}=\log a \)}[/tex]
ANSWER :-
[tex] \textsf{Let \( \bf u=x^{y} \) and \( \bf v=y^{x} \Rightarrow u+v=\log a \)}[/tex]
[tex] \pmb{ \[ \begin{array}{ll} \\ \\ \Rightarrow \quad &\displaystyle\sf \frac{d u}{d x}+\frac{d v}{d x}=0 \qquad\qquad\qquad\cdots\cdots (i)\\ \\ \\ &\displaystyle\sf \log u =y \log x \\ \\ \\ \Rightarrow \quad &\displaystyle\sf \frac{1}{ u } \frac{ du }{ d x}=\frac{y}{x}+\log x \frac{ dy }{ d x} \\ \\ \\ \Rightarrow \quad &\displaystyle\sf \frac{ du }{ d x}=x^{y}\left[\frac{y}{x}+\log x \frac{ dy }{ d x}\right] \\ \\ \\ & \displaystyle\sf \log v =x \log y \\ \\ \\ \Rightarrow \quad& \displaystyle\sf \frac{1}{ v } \frac{ dv }{ dx }=\frac{x}{y} \frac{d y}{ d x}+\log y \\ \\ \\ \Rightarrow \quad &\displaystyle\sf \frac{d v}{ d x}=y^{z}\left[\frac{x}{y} \frac{d y}{d x}+\log y\right] \end{array} \] } [/tex]
[tex]\pmb{ \begin{array}{ll} \\ \\ \Rightarrow & \displaystyle \sf y x^{y-1}+x^{y} \log x \frac{ d y}{ d x}+x y^{x-1} \frac{ d y}{ d x}+y^{x} \log y=0 \qquad\texttt{[From (i)]}\\ \\ \\ \blue\Rightarrow & \red{ \displaystyle\sf \frac{d y}{ d x}=-\left[\frac{y \cdot x^{y-1}+y^{x} \cdot \log y}{x y^{x-1}+x^{y} \cdot \log x}\right]}\end{array} }[/tex]
[tex]\\\\\rule{170pt}{0.1pt} \\ \\\small \fcolorbox{magenta}{lavender}{ \color{darkcyan} \textrm{Answerd by ★彡[@ᎥᏖᏕᎩᎧᏬᏒᎮᏗᎶᏝᏗ02]彡★}}[/tex]