[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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