If x^m yⁿ = 2(x + y)^{m + n} , find the value of \(\frac{\mathrm{d} y}{\mathrm{d} x} \)

Concept:

log ( a + b ) = log a + log b

 \(\frac{\mathrm{d} \log x}{\mathrm{d} x} = \frac{1}{\rm x}\)

Calculation:

 x^m yⁿ  = 2(x + y)^{m + n} 

Taking log on both sides , we get 

⇒ log(x^m 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 + \(\frac{\mathrm{d} \rm y}{\mathrm{d} x}\)]

⇒ \(\frac{\mathrm{d} \rm y}{\mathrm{d} x}\)\(\left ( \frac{\rm m + n}{\rm x + y} - \frac{\rm n}{\rm y} \right )\) = \(\frac{\rm m}{\rm x} - \frac{\rm m + n}{\rm (\rm x + y)}\) 

⇒ \(\frac{\mathrm{d} \rm y}{\mathrm{d} x}\) = \(\frac{\rm y}{\rm x}\)