# If \[y=\dfrac{\sin x}{1+\dfrac{\cos }{1+\dfrac{\sin x}{1+\dfrac{\cos x}{1+\sin x........}}}}\], then find \[\dfrac{dy}{dx}\].

Last updated date: 25th Mar 2023

•

Total views: 312.3k

•

Views today: 3.89k

Answer

Verified

312.3k+ views

Hint: Simplify the denominator which is an infinite function and differentiate by applying product rule.

Here \[y\] is an infinite function. By observation, we can see the pattern that \[\sin x\] and \[\cos x\]appear alternatingly and hence, we can write \[y\] as

\[y=\dfrac{\sin x}{1+\dfrac{\cos x}{1+y}}\]

\[\Rightarrow y=\dfrac{\sin x}{\dfrac{1+y+\cos x}{1+y}}\]

\[\Rightarrow y=\dfrac{\sin x(1+y)}{1+y+\cos x}\]

\[\Rightarrow y(1+y+\cos x)=\sin x(1+y)\]

Differentiating both sides with respect to\[x\], we get,

\[(1+y+\cos x).\dfrac{dy}{dx}+y(1+\dfrac{dy}{dx}-\sin x)=(1+y)\cos x+\sin x(\dfrac{dy}{dx})\]

\[\Rightarrow (1+y+\cos x).\dfrac{dy}{dx}+y+y\dfrac{dy}{dx}-y\sin x=(1+y)\cos x+\sin x(\dfrac{dy}{dx})\]

Taking all the terms with \[\dfrac{dy}{dx}\]to one side and other terms to the other side, we get,

\[\Rightarrow (1+y+\cos x).\dfrac{dy}{dx}+y\dfrac{dy}{dx}-\sin x(\dfrac{dy}{dx})=(1+y)\cos x+y\sin x-y\]

\[\Rightarrow (1+y+\cos x-\sin x+y).\dfrac{dy}{dx}=(1+y)\cos x+y\sin x-y\]

\[\Rightarrow \dfrac{dy}{dx}=\dfrac{y(cosx+\sin x-1)+\cos x}{1+2y+\cos x-\sin x}\]

Note: Generally students make a mistake of writing \[\dfrac{\sin x}{1+\dfrac{\cos }{1+\dfrac{sinx}{1+\cos x.....}}}\] as \[\dfrac{\sin x}{\dfrac{1+\cos x}{y}}\]which is wrong as the repetition starts after the addition of \[1\] in the denominator i.e. \[\dfrac{\sin x}{\dfrac{1+\cos x}{1+y}}\].

Here \[y\] is an infinite function. By observation, we can see the pattern that \[\sin x\] and \[\cos x\]appear alternatingly and hence, we can write \[y\] as

\[y=\dfrac{\sin x}{1+\dfrac{\cos x}{1+y}}\]

\[\Rightarrow y=\dfrac{\sin x}{\dfrac{1+y+\cos x}{1+y}}\]

\[\Rightarrow y=\dfrac{\sin x(1+y)}{1+y+\cos x}\]

\[\Rightarrow y(1+y+\cos x)=\sin x(1+y)\]

Differentiating both sides with respect to\[x\], we get,

\[(1+y+\cos x).\dfrac{dy}{dx}+y(1+\dfrac{dy}{dx}-\sin x)=(1+y)\cos x+\sin x(\dfrac{dy}{dx})\]

\[\Rightarrow (1+y+\cos x).\dfrac{dy}{dx}+y+y\dfrac{dy}{dx}-y\sin x=(1+y)\cos x+\sin x(\dfrac{dy}{dx})\]

Taking all the terms with \[\dfrac{dy}{dx}\]to one side and other terms to the other side, we get,

\[\Rightarrow (1+y+\cos x).\dfrac{dy}{dx}+y\dfrac{dy}{dx}-\sin x(\dfrac{dy}{dx})=(1+y)\cos x+y\sin x-y\]

\[\Rightarrow (1+y+\cos x-\sin x+y).\dfrac{dy}{dx}=(1+y)\cos x+y\sin x-y\]

\[\Rightarrow \dfrac{dy}{dx}=\dfrac{y(cosx+\sin x-1)+\cos x}{1+2y+\cos x-\sin x}\]

Note: Generally students make a mistake of writing \[\dfrac{\sin x}{1+\dfrac{\cos }{1+\dfrac{sinx}{1+\cos x.....}}}\] as \[\dfrac{\sin x}{\dfrac{1+\cos x}{y}}\]which is wrong as the repetition starts after the addition of \[1\] in the denominator i.e. \[\dfrac{\sin x}{\dfrac{1+\cos x}{1+y}}\].

Recently Updated Pages

Calculate the entropy change involved in the conversion class 11 chemistry JEE_Main

The law formulated by Dr Nernst is A First law of thermodynamics class 11 chemistry JEE_Main

For the reaction at rm0rm0rmC and normal pressure A class 11 chemistry JEE_Main

An engine operating between rm15rm0rm0rmCand rm2rm5rm0rmC class 11 chemistry JEE_Main

For the reaction rm2Clg to rmCrmlrm2rmg the signs of class 11 chemistry JEE_Main

The enthalpy change for the transition of liquid water class 11 chemistry JEE_Main

Trending doubts

Name the Largest and the Smallest Cell in the Human Body ?

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

Epipetalous and syngenesious stamens occur in aSolanaceae class 11 biology CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

A ball impinges directly on a similar ball at rest class 11 physics CBSE

Lysosomes are known as suicidal bags of cell why class 11 biology CBSE

Two balls are dropped from different heights at different class 11 physics CBSE

A 30 solution of H2O2 is marketed as 100 volume hydrogen class 11 chemistry JEE_Main

A sample of an ideal gas is expanded from 1dm3 to 3dm3 class 11 chemistry CBSE