/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 126 If \(y=\sin ^{-1} x\), prove tha... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 126

If \(y=\sin ^{-1} x\), prove that \(\left(1-x^{2}\right) y_{2}-x y_{1}=0\)

Short Answer

Expert verified
This exercise is a proof and the short answer is that we verified that indeed \(\left(1-x^{2}\right) y_{2}-x y_{1}=0\), where \(y_{1}\) and \(y_{2}\) represent the first and second derivatives respectively of the function \(y=\sin ^{-1} x\).

Step by step solution

01

Differentiate to find \(y'\)

The derivative of \(y=\sin ^{-1} x\) is known and can be obtained as \(y_{1}=y'=\frac{1}{\sqrt{1-x^{2}}}\) using the differentiation rules
02

Differentiate to find \(y''\)

Take the derivative of \(y'\) again using chain rule to find \(y''\). Before that, rewrite \(y'\) as \(y_{1}=\frac{1}{\sqrt{1-x^{2}}}=\frac{1}{\sqrt{(1-x)^{(1-x)}}}\), then when taking the derivative, the negative sign appears from the chain rule and the power drops by one. Therefore, \(y_{2}=y''=\frac{-x}{(1-x^{2})^{3/2}}\)
03

Show that \(\left(1-x^{2}\right) y_{2}-x y_{1}=0\)

Now, substituting \(y_{1}\) and \(y_{2}\) into the given expression gives \(\left(1-x^{2}\right)(-\frac{x}{(1-x^{2})^{3/2}})-x(\frac{1}{\sqrt{1-x^{2}}})\). This simplifies to \(-x+x^{3}-\frac{x}{\sqrt{1-x^2}}\). Because \(x^{2}\) under the root on the denominator of the second term, it simplifies to get \(-x+x^{3}-\frac{x(1-x^{2})}{1-x^{2}}=-x+x^{3}-x=-x+x^{3}-x=0\). Therefore, \(\left(1-x^{2}\right) y_{2}-x y_{1}=0\)

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