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# If log y = tan^{–1} X, show that : (1+x^{2})y_{2}+(2x–1) y_{1}=0.

Answer :

Note: y_{2} represents second order derivative i.e. and y_{1} = dy/dx

Given,

log y = tan^{–1} X

∴ y = ……equation 1

to prove : (1+x^{2})y_{2}+(2x–1)y_{1}=0

We notice a second order derivative in the expression to be proved so first take the step to find the second order derivative.

Let’s find

As

So, lets first find dy/dx

Using chain rule, we will differentiate the above expression

Let t = tan^{–1} x => []

And y = e^{t}

…….equation 2

Again differentiating with respect to x applying product rule:

Using chain rule we will differentiate the above expression-

[using & ]

Using equation 2 :

∴ (1+x^{2})y_{2}+(2x–1)y_{1}=0 ……proved

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