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Chapter 6: Problem 32

If \(y=\sqrt{x-1}+\sqrt{x+1}\), prove that \(\sqrt{x^{2}-1} \frac{d y}{d x}=\frac{1}{2} y\).

Short Answer

Expert verified
The given equation is proved by first deriving the function \(y\), substituting the derivative in place of \(\frac{dy}{dx}\) in the left-hand side of the equation, simplifying this side, and finally showing that both sides of the equation are indeed equal.

Step by step solution

01

Write down the given function

The given function is \(y=\sqrt{x-1}+\sqrt{x+1}\).
02

Derive the given function to obtain \(\frac{dy}{dx}\)

The derivative of \(y\) with respect to \(x\) can be calculated as such: \(\frac{dy}{dx}=\frac{1}{2(\sqrt{x-1})}-\frac{1}{2(\sqrt{x+1})}\) after using the chain rule in calculus.
03

Substitute the value of \(\frac{dy}{dx}\) into the left-hand side of the given equation

Substituting, we get \(\sqrt{x^{2}-1}*\frac{dy}{dx} = \sqrt{x^2-1} * (\frac{1}{2(\sqrt{x-1})}-\frac{1}{2(\sqrt{x+1})})\). Simplifying, we get \(\frac{\sqrt{x^2-1}}{2(\sqrt{x-1})} - \frac{\sqrt{x^2-1}}{2(\sqrt{x+1})}\). In algebra, we can express \(\sqrt{x^2-1} = \sqrt{x-1}*\sqrt{x+1}\). Substituting this into the equation gives us \(\frac{\sqrt{x-1}*\sqrt{x+1}}{2(\sqrt{x-1})} - \frac{\sqrt{x-1}*\sqrt{x+1}}{2(\sqrt{x+1})}\) which simplifies to \(\frac{\sqrt{x+1}}{2} + \frac{\sqrt{x-1}}{2}\).
04

Compare the simplified left-hand side to the right-hand side of the given equation

Seeing as \(\frac{\sqrt{x+1}}{2} + \frac{\sqrt{x-1}}{2} = \frac{1}{2} y\), it can be confirmed that the two sides are identical, thereby proving the given equation.

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Most popular questions from this chapter

If \(\tan ^{-1}\left(\frac{x^{2}-y^{2}}{x^{2}+y^{2}}\right)=a\) prove that \(\frac{d y}{d x}=\frac{x(1-\tan a)}{y(1+\tan a)}\).

If \(e^{x+y}-x=0\), prove that \(\frac{d y}{d x}=\frac{1-x}{x}\).

If \(y=\sqrt{\cos x+\sqrt{\cos x+\sqrt{\cos x+\ldots \text { to } \infty}}}\) then find \(\frac{d y}{d x}\).

If \(y=\sin ^{-1}\left(x \sqrt{1-x}-\sqrt{x-x^{3}}\right)\), find \(\frac{d y}{d x}\)

If \(y=\sqrt{x+\sqrt{x}+\sqrt{x}+\ldots}\) to \(\infty\), prove that \(\frac{d y}{d x}=\frac{1}{2 y-1}\)
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