Question

d²x/dy²=.........,Select Proper option from the given options.
(a) 1/d²y/dx²
(b) 1/(dy/dx)²
(c) - 1/(dy/dx)²
(d) - 1/(dy/dx)³ d²y/dx²

Answer 1

option a) is the correct 1/d^2y/dx^2

Answer 2

we have to get value of d²x/dy² in in given options.

$\frac{dx}{dy}=\frac{1}{\frac{dy}{dx}}$

now differentiate both sides,
$\frac{d^2x}{dy^2}=\frac{d}{dy}\left(\frac{dx}{dy}\right)=\frac{d}{dy}\left(\frac{1}{\frac{dy}{dx}}\right)$

now by quotient rule , we can say that ,

$\begin{aligned}\frac{d}{dy}\left(\frac{1}{\dfrac{dy}{dx}}\right) &= \frac{-\dfrac{d}{dy}\left(\dfrac{dy}{dx}\right)}{\left(\dfrac{dy}{dx}\right)^2}\\ &= -\frac{\dfrac{dx}{dy}\cdot\dfrac{d}{dx}\left(\dfrac{dy}{dx}\right)}{\left(\dfrac{dy}{dx}\right)^2} \\ &= -\dfrac{\dfrac{dx}{dy}\cdot\dfrac{d^2y}{dx^2}}{\left(\dfrac{dy}{dx}\right)^2} \\ &= -\frac{\dfrac{d^2y}{dx^2}}{\left(\dfrac{dy}{dx}\right)^3}\end{aligned}$

hence, option (d) is correct.