Continuity and ...

If $$y = x - x^{2}$$, then the derivative of $$y^{2} w.r.t. x^{2}$$ is

Find the derivative of $$\sin(2\sin^{-1}x)$$.

Let $$y=\sin^{-1}x$$, then find $$(1-x^{2})y_{2}-xy_{1}$$.
Where $$y_{1}$$ and $$y_{2}$$ denote the first and second order derivatives respectively.

Differentiate $$\cos^{-1}(4x^{3}-3x)$$ w.r.t $$x$$.

Find derivative of $$\tan^{-1}\dfrac{\cos x}{1+\sin x}$$.

If $$y=\dfrac{1}{2}(\sin^{-1}x)^{2}$$, then find $$(1-x^{2})y_{2}-xy_{1}$$. 
Where $$y_{1}$$ and $$y_{2}$$ denote first and second derivatives of $$y$$ respectively.

The derivative of $$\sin^{-1}x$$ with respect to $$\cos^{-1}\sqrt{1-x^2}$$ is?

If $$y=\sin^{-1}(x^{2})$$ then find $$\dfrac{dy}{dx}$$ using first principle.

If $$y=\cos^{-1}(\sqrt{x})$$, then find $$\dfrac{dy}{dx}$$ using first principle.

Find derivative of $$\sin^{-1}(x^{2})$$ using first principle.