Question

find derivative by using multiplying by log method.​

Answer 1

❏Question :-

Find the derivative by using log .

$\to \bf y = x + \frac{1}{x + \frac{1}{x + \frac{1}{x + } ....} }$

❏Answer :-

$\boxed{ \to \bf \red{ \frac{dy}{dx} = } \green{ \frac{y}{xy + 2} }} \:$

❏Step - by - step explanation :-

We have ,

$\leadsto \bf y \red{ = x + }\frac{1}{x + \green{ \frac{1}{x + \frac{1}{x + }} ....} } \: \\ \\ \: \mathfrak{ \blue{we \: can \: wite \: it \: }} \\ \\ \to \bf y = x + \frac{1}{y } \\ \bf \purple{ taking \: \log \: on \: both \: sides }\: \\ \\ \to \bf \: log(y) = \blue{ log \bigg(x + \frac{1}{y} \bigg) }\\ \\ \sf now \: differentiate \: with \: respect \: to \: \bf{x }\\ \\ \to \bf \orange{\frac{1}{y} }\red{ \frac{dy}{dx} } = \red{ \frac{1}{x + \frac{1}{y} } } \green{ \frac{d}{dx} \bigg(x + \frac{1}{y} \bigg)} \\ \\ \to \bf \red{\frac{1}{y} }\green{ \frac{dy}{dx} } = \: \frac{1}{ \frac{xy + 1}{y} } \bigg(1 \blue{- \frac{1}{ {y}^{2} } \frac{dy}{dx} \bigg)} \\ \\ \to \bf \orange{\frac{1}{y} }\red{ \frac{dy}{dx} } = \frac{y}{xy + 1} - \bigg( \red{\frac{y}{xy + 1} }\bigg) \bigg( \green{ \frac{1}{ {y}^{2} } \frac{dy}{dx} } \bigg) \\ \\ \to \bf \red{\frac{1}{y} } \pink{\frac{dy}{dx} } = \frac{y}{xy + 1} \blue{ - \frac{1}{y(xy + 1)}} \frac{dy}{dx} \\ \\ \to \bf \red{ \frac{1}{y} } \frac{dy}{dx} \green{ + \frac{1}{y(xy + 1)} } \frac{dy}{dx} = \purple{ \frac{y}{xy + 1} }\\ \\ \to \bf \frac{dy}{dx} \bigg( \pink{ \frac{1}{y} } + \orange{ \frac{1}{y(xy + 1)} } \bigg) = \blue{ \frac{1}{xy + 1} }\\ \\ \to \bf \frac{dy}{dx} \bigg( \green{ \frac{xy + 1 + 1}{y(xy + 1)}} \bigg) = \pink{\frac{1}{xy + 1} } \\ \\ \boxed{ \to \bf \red{ \frac{dy}{dx} = } \green{ \frac{y}{xy + 2} }}$