# Differentiation of Infinite Series Class 12

Here you will learn what is differentiation of infinite series class 12 with examples.

Let’s begin –

## Differentiation of Infinite Series

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

Example 1 : If y = $$x^{x^{x^{…\infty}}}$$, find $$dy\over dx$$.

Solution : Since by deleting a single term from an infinite series, it remains same. Therefore, the given function may be written as

y = $$x^y$$

Taking log on both sides,

$$\implies$$ log y = y logx

Differentiating both sides with respect to x,

$$1\over y$$$$dy\over dx$$ = $$dy\over dx$$ log x + y $$d\over dx$$ (log x)

$$1\over y$$$$dy\over dx$$ = $$dy\over dx$$ log x + $$y\over x$$

$$dy\over dx$${$${{1\over y} – log x}$$} = $$y\over x$$

$$\implies$$ $$dy\over dx$$$$(1 – y log x)\over y$$ = $$y\over x$$

$$\implies$$ $$dy\over dx$$ = $$y^2\over {x(1 – ylog x)}$$

Example 2 : If y = $$\sqrt{sinx + \sqrt{sinx + \sqrt{sinx + ……. to \infty}}}$$, find $$dy\over dx$$.

Solution : Since by deleting a single term from an infinite series, it remains same. Therefore, the given function may be written as

y = $$\sqrt{sin x + y}$$

Squaring on both sides,

$$\implies$$  $$y^2$$  = sin x + y

Differentiating both sides with respect to x,

2y $$dy\over dx$$ =cosx +  $$dy\over dx$$

$$\implies$$ $$dy\over dx$$$$(2y – 1)$$ = cos x

$$\implies$$ $$dy\over dx$$ = $$cos x\over {2y – 1}$$