Find the derivative of $e^{x}+e^{y}=e^{x+y}$.

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

If $\log (x+y)-2xy=0$,then $y^{\prime}(0)=$

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

$f(x)$ and $g(x)$ are two differentiable functions on $[0, 2]$ such that $f''(x) - g''(x) = 0$,$f'(1) = 2$,$g'(1) = 4$,$f(2) = 3$,and $g(2) = 9$. Then $f(x) - g(x)$ at $x = 3/2$ is:

Let $f$ be a differentiable function satisfying $f(x + 2y) = 2yf(x) + xf(y) - 3xy + 1$ for all $x, y \in R$ such that $f'(0) = 1$. Then $f(2)$ is equal to: