Question

If $ x^2y^5 = (x + y)^7 $ , then $ \frac{d^{2}y}{dx^{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{d y}{d x}=\frac{7}{x+y}\left(1+\frac{d y}{d x}\right) $
$\Rightarrow \frac{d y}{d x}\left(\frac{7}{x+y}-\frac{5}{y}\right)=\frac{2}{x}-\frac{7}{x+y}$
$\Rightarrow \frac{d y}{d x}=\frac{y}{x} \,\,\,\,...(i)$
Again, differentiating, we get
$\frac{d^{2} y}{d x^{2}} =\frac{x \frac{d y}{d x}-y}{x^{2}} $
$=\frac{x \cdot(y / x)-y}{x^{2}}\,\,\,\, $[ { from Eq.(i) }
$=0$