If $\tan^{-1}(x^2 + y^2) = \alpha$,then $\frac{dy}{dx}$ is equal to

Find $\frac{dy}{dx}$ for the equation $\sin^{2} x + \cos^{2} y = 1$.

Let $f: R \rightarrow R$ be a differentiable function and $f(1)=4$. Then the value of $\lim _{x \rightarrow 1} \int_4^{f(x)} \frac{2 t}{x-1} dt$, if $f^{\prime}(1)=2$ is

Find $\frac{dy}{dx}$ for the function $y^{x} = x^{y}$.

If $x = \exp \left\{ {{{\tan }^{ - 1}}\left( {{{y - {x^2}} \over {{x^2}}}} \right)} \right\}$,then $\frac{dy}{dx}$ equals

If $2x^2 + 3xy - y^2 + 4x - 5y + 6 = 0$, then the value of $\frac{dy}{dx}$ at $(x, y) = (1, -2)$ is