For the curve $\sqrt{x} + \sqrt{y} = 1$,the value of $\frac{dy}{dx}$ at the point $\left( \frac{1}{4}, \frac{1}{4} \right)$ is:

Let $f$ be a differentiable function such that $(x - y) f(x + y) - (x + y) f(x - y) = 4xy(x^2 - y^2)$ and $f(1) = 2$. Then the area enclosed by $\frac{|f(x) - x|^{1/3}}{17} + \frac{|f(y) - y|^{1/3}}{2} \le \frac{1}{4}$ is

If $f(1)=3$ and $f^{\prime}(1)=2$,then the value of $\frac{d}{d x}\left\{\log \left[f\left(e^x+2 x\right)\right]\right\}$ at $x=0$ is

If $y=\sqrt{(x-\sin x)+\sqrt{(x-\sin x)+\sqrt{(x-\sin x) \ldots}}}$,then $\frac{dy}{dx}=$

If $\ln \left( {(e - 1){e^{xy}} + {x^2}} \right) = {x^2} + {y^2}$,then $\left. {\frac{{dy}}{{dx}}} \right|_{(1,0)}$ is

If $3 f(\cos x) + 2 f(\sin x) = 5 x$,then $f^{\prime}(\cos x) + f^{\prime}(\sin x) =$