Math, asked by bhjkkhtr, 8 months ago

if y=tan(x+y) ,find dy/dx​

Answers

Answered by shadowsabers03
4

Given,

\longrightarrow y=\tan(x+y)

Taking tan inverse on both sides,

\longrightarrow \tan^{-1}y=x+y

\longrightarrow x=\tan^{-1}y-y

Differentiating with respect to y,

\longrightarrow \dfrac{dx}{dy}=\dfrac{d}{dy}\left[\tan^{-1}y-y\right]

We have,

  • \dfrac{d}{dy}\big(\tan^{-1}y\big)=\dfrac{1}{1+y^2}

\longrightarrow \dfrac{d}{dy}\big(y\big)=1

Then,

\longrightarrow \dfrac{dx}{dy}=\dfrac{1}{1+y^2}-1

\longrightarrow \dfrac{dx}{dy}=\dfrac{1-(1+y^2)}{1+y^2}

\longrightarrow \dfrac{dx}{dy}=\dfrac{1-1-y^2}{1+y^2}

\longrightarrow \dfrac{dx}{dy}=-\dfrac{y^2}{1+y^2}

Taking the reciprocal we get,

\longrightarrow \dfrac{dy}{dx}=-\dfrac{1+y^2}{y^2}

\longrightarrow \underline{\underline{\dfrac{dy}{dx}=-1-\dfrac{1}{y^2}}}

This is the derivative.

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