Question

$\rm If \: u = lx + my , v= mx - ly \: then \\ \rm show \: that \: \bigg( \frac{\partial u}{\partial x} \bigg ) \bigg( \frac{\partial x}{\partial u} \bigg)= \frac{ {l}^{2} }{ {l}^{2} } +m^2$

​

Answer 1

Appropriate Question :-

$If \: u = lx + my , v= mx - ly \: then \\ show \: that \: \bigg( \frac{\partial u}{\partial x} \bigg ) \bigg( \frac{\partial x}{\partial u} \bigg)= \frac{ {l}^{2} }{ {l}^{2} + {m}^{2}}$

$\large\underline{\sf{Solution-}}$

Given that,

$\: u = lx + my - - - (1)$

On differentiating both sides partially w. r. t. x, we get

$\implies \:\dfrac{\partial u}{\partial x} = l - - - (i)$

Now, further given that

$v = mx - ly - - - (2)$

On multiply equation (1) by l and (2) by m, we get

$vm = {m}^{2} x - lmy - - - (3)$

and

$\: ul = {l}^{2} x + mly - - - (4)$

On adding equation (3) and (4), we get

${l}^{2}x + {m}^{2}x= lu + mv$

$({l}^{2} + {m}^{2})x= lu + mv$

On differentiating both sides partially w. r. t. u, we get

$({l}^{2} + {m}^{2})\frac{\partial }{\partial u}x=\frac{\partial }{\partial u} (lu + mv)$

$({l}^{2} + {m}^{2})\frac{\partial x}{\partial u}=l$

$\implies \:\frac{\partial x}{\partial u}=\dfrac{l}{{l}^{2} + {m}^{2}} - - - (5)$

Now, Consider

$\rm \: \dfrac{\partial u}{\partial x} \times \dfrac{\partial x}{\partial u}$

On substituting the values from equation (i) and (5), we get

$= l \times \dfrac{l}{{l}^{2} + {m}^{2}}$

$= \dfrac{ {l}^{2} }{{l}^{2} + {m}^{2}}$

Hence,

$\implies \:\boxed{{ \: \:\bigg(\frac{\partial u}{\partial x}\bigg) \bigg(\frac{\partial x}{\partial u}\bigg)= \dfrac{ {l}^{2} }{{l}^{2} + {m}^{2}} \: \: \: }}$

$\rule{190pt}{2pt}$

Additional information :-

$\begin{gathered}\boxed{\begin{array}{c|c} \bf f(x) & \bf \dfrac{d}{dx}f(x) \\ \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf k & \sf 0 \\ \\ \sf sinx & \sf cosx \\ \\ \sf cosx & \sf - \: sinx \\ \\ \sf tanx & \sf {sec}^{2}x \\ \\ \sf cotx & \sf - {cosec}^{2}x \\ \\ \sf secx & \sf secx \: tanx\\ \\ \sf cosecx & \sf - \: cosecx \: cotx\\ \\ \sf \sqrt{x} & \sf \dfrac{1}{2 \sqrt{x} } \\ \\ \sf logx & \sf \dfrac{1}{x}\\ \\ \sf {e}^{x} & \sf {x}^{n}\\ \\ \sf {nx}^{n - 1} & \sf {e}^{x} \end{array}} \\ \end{gathered}$