Question

please help me do this​

Answer 1

Question :

If  $\sf \dfrac{dy}{dx}=\dfrac{1}{\frac{dx}{dy}}$ , Show that  $\sf \dfrac{d^2y}{dx^2}=\dfrac{-\frac{d^2x}{dy^2}}{\big(\frac{dx}{dy}\big)^3}$  

Solution :

LHS

$:\to \bf \dfrac{d^2y}{dx^2}$

$:\to \sf \dfrac{d}{dx}\bigg(\dfrac{dy}{dx}\bigg)$

$\bullet\ \; \; \sf \dfrac{dy}{dx}=\dfrac{1}{\frac{dx}{dy}}$

$:\to \sf \dfrac{d}{dx}\Bigg(\dfrac{1}{\frac{dx}{dy}}\Bigg)$

• Divide num. and den. by dy

$:\to \sf \dfrac{\frac{d}{dy}\bigg(\frac{1}{\frac{dx}{dy}}\bigg)}{\frac{dx}{dy}}$

$:\to \sf \dfrac{dy}{dx}\times \dfrac{d}{dy}\Bigg(\dfrac{1}{\frac{dx}{dy}}\Bigg)$

$\bullet\ \; \sf \dfrac{d}{dx}\bigg( \dfrac{u}{v}\bigg)=\dfrac{v\ u'-u\ v'}{v^2}$

$:\to \sf \dfrac{dy}{dx}\times \dfrac{\frac{dx}{dy}\times \frac{d}{dx}(1)-(1)\frac{d}{dy}\big(\frac{dx}{dy}\big)}{\big(\frac{dx}{dy}\big)^2}$

$:\to \sf \dfrac{dy}{dx}\times \dfrac{-\frac{d^2x}{dy^2}}{\big(\frac{dx}{dy}\big)^2}$

$:\leadsto \bf \pink{\dfrac{-\frac{d^2x}{dy^2}}{\big(\frac{dx}{dy}\big)^3}}\ \; \bigstar$

RHS