Question

Let $x^2{y}^3=(x+y{)}^5$ then find $\frac{dy}{dx}$

Answer 1

Given,

$x^2{y}^3=(x+y{)}^5$

Now taking logarithm with respect to the base $e$ both sides we get,

$2\log x+3\log y=5\log (x+y)$

Now differentiating both sides with respect to $x$ we get,

$\frac{2}{x}+\frac{3}{y}\frac{dy}{dx}=\frac{5}{x+y}(1+\frac{dy}{dx})$

or, $(\frac{2}{x}-\frac{5}{x+y})=(\frac{5}{x+y}-\frac{3}{y})\frac{dy}{dx}$

or, $\left(\frac{2y-3x}{x(x+y)}\right)=\left(\frac{2y-3x}{y(x+y)}\right)\frac{dy}{dx}$

or, $\frac{dy}{dx}=\frac{y}{x}$.