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