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Chapter 6: Problem 36
If \(y=\sqrt{x^{2}+a^{2}}\), prove that \(y \frac{d y}{d x}-x=0\).
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If \(y=\tan ^{-1} x\), then prove that (i) \(\left(1+x^{2}\right) y_{2}+2 x y_{1}=0\) (ii) \(\left(1+x^{2}\right) y_{n+2}+2(n+2) x y_{n+1}+n(n+1) y_{n}\) \(=0 .\)
\begin{aligned}
&\text { If } y=\sin ^{-1}\left(\frac{2 x}{1+x^{2}}\right)+\cos
^{-1}\left(\frac{1-x^{2}}{1+x^{2}}\right) \\
&0
If \(y=\frac{a x+b}{x^{2}+c}\), then show that $$ \left(2 x \frac{d y}{d x}+y\right) \frac{d^{3} y}{d x^{3}}=3\left(x \frac{d^{2} y}{d x^{2}}+\frac{d y}{d x}\right) \frac{d^{2} y}{d x^{2}} $$
Find \(\frac{d y}{d x}\), if \(y=x^{\sin x}\)
\begin{aligned} &\text { If } f(x)=x+\tan x \text { and } g \text { is the inverse of } f \text {, then }\\\ &\text { prove that } g^{\prime}(x)=\frac{1}{2+\tan ^{2}(g(x))} . \end{aligned}
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