If $\sin(xy) + \cos(xy) = 0$,then $\frac{dy}{dx} = $

The point which lies on the tangent drawn to the curve $x^4 e^y + 2 \sqrt{y+1} = 3$ at the point $(1,0)$ is

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

If $x^2+y^2=t+\frac{1}{t}$ and $x^4+y^4=t^2+\frac{1}{t^2}$,then $\frac{dy}{dx}$ is equal to

If $\sin (x+y)+\cos (x+y)=\sin \left[\cos ^{-1}\left(\frac{1}{3}\right)\right]$,then $\frac{d y}{d x}=$

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