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Chapter 6: Problem 70
Find \(\frac{d y}{d x}\), if \(y=x^{\sin x}\)
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The length of the longest interval in which Rolle's theorem can be applied for the function \(f(x)=\left|x^{2}-a^{2}\right|\) is \((a>0)\) (a) \(2 a\) (b) \(4 a^{2}\) (c) \(a \sqrt{2}\) (d) \(a\)
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 \(x=\sin ^{-1}\left(\frac{2 t}{1+t^{2}}\right)\) and \(y=\tan ^{-1}\left(\frac{2 t}{1-t^{2}}\right)\), \(t>1\), then prove that \(\frac{d x}{d y}=-1\)
If \(y=\frac{a x+b}{x^{2}+c}\), then show that $$ \left(2 x \frac{d y}{d x}+y\right) \frac{d^{3} y}{d x^{3}}=3\left(x \frac{d^{2} y}{d x^{2}}+\frac{d y}{d x}\right) \frac{d^{2} y}{d x^{2}} $$
If \((a+b x) e^{\frac{x}{y}}=x\), then prove that \(x^{3} \frac{d^{2} y}{d x^{2}}=\left(x \frac{d y}{d x}-y\right)^{2}\).
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