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Chapter 6: Problem 14
If \(y=\cos ^{-1}\left(\sqrt{\frac{\cos 3 x}{\cos ^{3} x}}\right)\), prove that \(\frac{d y}{d x}=\sqrt{\frac{6}{\cos 2 x+\cos 4 x}}\)
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If \(y=\left(1+\frac{1}{x}\right)^{x}+x^{\left(1+\frac{1}{x}\right)}\), find \(\frac{d y}{d x}\) at \(x=1\)
If \(\int_{0}^{1} e^{x}(x-\alpha) d x=0\), then (a) \(1<\alpha<2\) (b) \(\alpha \leq 2\) (c) \(0<\alpha<1\) (d) \(\alpha=0\)
\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 \(y=\cos ^{-1}\left(\frac{5 t+12 \sqrt{1-t^{2}}}{13}\right)\) and \(x=\cos ^{-1}\left(\frac{1-t^{2}}{1+t^{2}}\right)\), find \(\frac{d y}{d x}\)
If \(y=\sqrt{\cos x+\sqrt{\cos x+\sqrt{\cos x+\ldots \text { to } \infty}}}\) then find \(\frac{d y}{d x}\).
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