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Question
If \[\left| x \right| < 1 \text{ and y} = 1 + x + x^2 + . . \] to ∞, then find the value of \[\frac{dy}{dx}\] ?
Solution
\[\text{ We have, y }= 1 + x + x^2 + . . . . to \ \infty \]
\[ \Rightarrow y = \frac{1}{1 - x} ...........[\because \text{ It is a G.P with first term }1\text{ and common ratio } x]\]
\[\Rightarrow \frac{dy}{dx} = \frac{d}{dx}\left( \frac{1}{1 - x} \right)\]
\[ \Rightarrow \frac{dy}{dx} = - \frac{1}{\left( 1 - x \right)^2}\frac{d}{dx}\left( 1 - x \right)\]
\[ \Rightarrow \frac{dy}{dx} = - \frac{1}{\left( 1 - x \right)^2}\left( - 1 \right)\]
\[ \Rightarrow \frac{dy}{dx} = \frac{1}{\left( 1 - x \right)^2}\]
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