Find \({d^2y\over dx^2}\) if x² + y² = 4.

Find the derivatives of the following : y = x^cosx

Find the derivatives of the following :  \(\sqrt{xy}=e^{(x-y)}\)

Find the derivative of the tan (x + y) + tan (x - y) = x

Find the derivatives of the following : \(tan^{-1}\sqrt{1-cos \ x \over 1+ cos \ x}\)

Find the derivatives of the following : \(cos(2tan^{-1}\sqrt{1-x\over 1+x})\) .

Find the derivative with \(\left(\frac{\sin x}{1+\cos x}\right)\) with respect to \(\left(\frac{\cos x}{1+\sin x}\right)\).

If \(y={sin^{-1}x\over \sqrt{1-x^2}}\) , Show that (1 - x²) y₂ - 3x y₁ - y = 0.

If sin y = x sin (a + y), then prove that \({dy\over dx}={sin^2(a+y)\over sin \ a}, a\neq n \pi.\)

If y = \((cos^{-1}x)^2\) ,prove that \((1-x^2){d^2y\over dx^2}-x{dy\over dx}-2=0.\)  Hence find y₂ when x = 0