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

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

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