Question

If \[\left| x \right| < 1 \text{ and y} = 1 + x + x^2 + . . \]  to ∞, then find the value of  \[\frac{dy}{dx}\] ?

Answer 1

\[\text{ We have, y }= 1 + x + x^2 + . . . . to \ \infty \]

\[ \Rightarrow y = \frac{1}{1 - x} ...........[\because \text{ It is a G.P with first term }1\text{ and common ratio } x]\]

\[\Rightarrow \frac{dy}{dx} = \frac{d}{dx}\left( \frac{1}{1 - x} \right)\]

\[ \Rightarrow \frac{dy}{dx} = - \frac{1}{\left( 1 - x \right)^2}\frac{d}{dx}\left( 1 - x \right)\]

\[ \Rightarrow \frac{dy}{dx} = - \frac{1}{\left( 1 - x \right)^2}\left( - 1 \right)\]

\[ \Rightarrow \frac{dy}{dx} = \frac{1}{\left( 1 - x \right)^2}\]