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Chapter 6: Problem 19
If \(y=\log (\sqrt{x-1}-\sqrt{x+1})\), find \(\frac{d y}{d x}\)
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If \(y=x^{n-1} \ln x\), then prove that $$ x^{2}\left(\frac{d^{2} y}{d x^{2}}\right)+(3-2 n) x \frac{d y}{d x}+(n-1)^{2} y=0 $$
If \(y=\sin ^{-1}\left(\frac{x}{\sqrt{1+x^{2}}}\right)+\cos
^{-1}\left(\frac{1}{\sqrt{x^{2}+1}}\right)\)
\(0
If \(y=x+\tan x\), prove that \(\cos ^{2} x \frac{d^{2} y}{d x^{2}}-2 y+2 x=0\)
If \(y=e^{\tan ^{-1}} x\), show that $$ \left(x^{2}+1\right) \frac{d^{2} y}{d x^{2}}+(2 x-1) \frac{d y}{d x}=0 $$
If \(x=a\left(t+\frac{1}{t}\right)\) and \(y=a\left(t-\frac{1}{t}\right)\), then prove that \(\frac{d y}{d x}=\frac{x}{y}\)
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