Differentiation of Infinite series

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

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

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