[Solved] If Xm Yn = 2(x + Y)m + N , Find The Val
Concept:
log ( a + b ) = log a + log b
\(\frac{\mathrm{d} \log x}{\mathrm{d} x} = \frac{1}{\rm x}\)
Calculation:
xm yn = 2(x + y)m + n
Taking log on both sides , we get
⇒ log(xm yn) = log[2(x + y)m + n]
⇒ m log x + n log y = log 2 + (m + n) log (x + y)
on differentiating both sides with respect to x , we get
⇒ \(\frac{\rm m}{\rm x}\) + \(\frac{\rm n}{\rm y}\frac{\mathrm{d} y}{\mathrm{d} x}\) = \(\frac{\rm m + n}{\rm x + y}\) [1 + \(\frac{\mathrm{d} \rm y}{\mathrm{d} x}\)]
⇒ \(\frac{\mathrm{d} \rm y}{\mathrm{d} x}\)\(\left ( \frac{\rm m + n}{\rm x + y} - \frac{\rm n}{\rm y} \right )\) = \(\frac{\rm m}{\rm x} - \frac{\rm m + n}{\rm (\rm x + y)}\)
⇒ \(\frac{\mathrm{d} \rm y}{\mathrm{d} x}\) = \(\frac{\rm y}{\rm x}\)
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