i am stuck in the end...one term does not match. otherwise the answer wud come...

question is:

if $\displaystyle x^p*y^q=(x+y)^{p+q}$ then prove that dy/dx =y/x and d/dx(dy/dx)=0.

please help

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- Jul 10th 2009, 01:11 AMdarachit[SOLVED] differentiation...
i am stuck in the end...one term does not match. otherwise the answer wud come...

question is:

if $\displaystyle x^p*y^q=(x+y)^{p+q}$ then prove that dy/dx =y/x and d/dx(dy/dx)=0.

please help - Jul 10th 2009, 01:51 AMGrandad
Hello darachitTake logs of both sides:

$\displaystyle p\ln x + q \ln y = (p+q)\ln(x+y)$

$\displaystyle \Rightarrow \frac{p}{x}+ \frac{q}{y}\frac{dy}{dx}=\frac{p+q}{x+y}\left(1+\f rac{dy}{dx}\right)$

If you now solve for $\displaystyle \frac{dy}{dx}$ and simplify, the equation becomes

$\displaystyle \frac{dy}{dx}\Big(x(qx-py)\Big)= y(qx-py)$

$\displaystyle \Rightarrow \frac{dy}{dx}=\frac{y}{x}$

Then $\displaystyle \frac{d}{dx}\Big(\frac{dy}{dx}\Big) = \frac{x\frac{dy}{dx}-y}{x^2}= 0$

Grandad - Jul 10th 2009, 02:08 AMdarachit
thanx ALOT!