Ucale

# Differentiation of Infinite series

Â If f (x) is represented by theÂ sumÂ of a powerÂ series. with radius of convergence r > 0 and – r < x < r, then the function has the infinite Â derivative or infinite differentiation.

Sometimes the value of y is given as an infinite series and we are asked to findÂ $\displaystyle \frac { dy }{ dx }$. In such cases we use the fact that if a term is deleted from an infinite series, it remains unaffected.The method of findingÂ  $\displaystyle \frac { dy }{ dx }$. is explained in the following examples.

### Example

$\displaystyle If\quad y=\sqrt { \sin { x } +\sqrt { \sin { x } \sqrt { \sin { x } } } ........\infty } ,\quad prove\quad that\quad \frac { dy }{ dx } =\frac { \cos { x } }{ 2y-1 }$

Solution:

The given series may be written as

$\displaystyle y=\sqrt { \sin { x } +y } \\ \Rightarrow { y }^{ 2 }=\sin { x } +y$

Differentiating both sides w.r.t. x, we get

$\displaystyle 2y\frac { dy }{ dx } =\cos { x } +\frac { dy }{ dx } \\ \Rightarrow \frac { dy }{ dx } \left( 2y-1 \right) =\cos { x } \\ \Rightarrow \frac { dy }{ dx } =\frac { \cos { x } }{ \left( 2y-1 \right) }$

April 18, 2019
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