If $\sqrt{\frac{y}{x}}+\sqrt{\frac{x}{y}}=2$, then $\frac{dy}{dx}$ is equal to

If $\tan y = \cot \left(\frac{\pi}{4} - x\right)$, then $\frac{dy}{dx} = $

Let $f: R \rightarrow R$ be a differentiable function with $f(0)=1$ and satisfying the equation $f(x+y)=f(x) \cdot f^{\prime}(y)+f^{\prime}(x) \cdot f(y)$ for all $x, y \in R$. Then the value of $\log (f(4))$ is

If $x e^{xy} = y + \sin^2 x$, then $\frac{dy}{dx}$ at $x = 0$ is

If $x \sqrt{1+y}+y \sqrt{1+x}=0$,then $\frac{d y}{d x}$ is equal to

If $y=y(x)$ and it follows the relation $4x{e^{xy}} = y + 5{\sin ^2}x$,then $y'(0)$ is equal to