Question

If $x^2{y}^5=(x+y{)}^7$ , then $\frac{d^{2}y}{d{x}^2}$ is equal to

• A. $y/x^2$
• B. $x/y$
• C. $1$
• D. $0$

Answer 1

Answer: (D)

Given, $x^2{y}^5=(x+y{)}^7$
Taking log on both sides, we get
$2\log x+5\log y=7\log (x+y)$
On differentiating, we get
$\frac{2}{x}+\frac{5}{y}\frac{dy}{dx}=\frac{7}{x+y}\left(1+\frac{dy}{dx}\right)$
$\Rightarrow \frac{dy}{dx}\left(\frac{7}{x+y}-\frac{5}{y}\right)=\frac{2}{x}-\frac{7}{x+y}$
$\Rightarrow \frac{dy}{dx}=\frac{y}{x}...(i)$
Again, differentiating, we get
$\frac{d^{2}y}{d{x}^2}=\frac{x\frac{dy}{dx}-y}{x^2}$
$=\frac{x\cdot (y/x)-y}{x^2}$[ { from Eq.(i) }
$=0$