Question

if xm yn = (x+y)m+n show that dy/dx= y/x and find d2y/dx2 plzzz help !!

Answer 1

$\text{I have applied logarithmic differentition to find}$

$\text{the derivative of the given function}$

$\textbf{Given:}$

$x^my^n=(x+y)^{m+n}$

$\text{Take logarithm on both sides}$

$m\;log\;x+n\;log\;y=(m+n)\;log\;(x+y)$

$\text{Differentiate with respect to x}$

$\displaystyle\frac{m}{x}+\frac{n}{y}\;\frac{dy}{dx}=\frac{m+n}{x+y}(1+\frac{dy}{dx})$

$\implies\displaystyle\frac{m}{x}+\frac{n}{y}\;\frac{dy}{dx}=\frac{m+n}{x+y}+\frac{m+n}{x+y}\frac{dy}{dx}$

$\implies\displaystyle(\frac{n}{y}-\frac{m+n}{x+y})\frac{dy}{dx}=\frac{m+n}{x+y}-\frac{m}{x}$

$\implies\displaystyle(\frac{nx+ny-my-ny}{y(x+y)})\frac{dy}{dx}=\frac{mx+nx-mx-my}{x(x+y)}$

$\implies\displaystyle(\frac{nx-my}{y})\frac{dy}{dx}=\frac{nx-my}{x}$

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

$\text{Differentiate again with respect to x}$

$\text{Using quotient rule of logarithm}$

$\boxed{\bf\frac{d(\frac{u}{v})}{dx}=\frac{v\,\frac{du}{dx}-u\,\frac{dv}{dx}}{v^2}}$

$\implies\displaystyle\frac{d^2y}{dx^2}&=\frac{x\times\frac{dy}{dx}-y\times1}{x^2}$

$&=\displaystyle\frac{x\times\frac{y}{x}-y}{x^2}$

$&=0$