If $x^3 - 2x^2y^2 + 5x + y - 5 = 0$, then at $(1, 1)$, $y''_1 = $?

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

If $\sin x \sqrt{\cos y} - \cos y \sqrt{\sin x} = 0$,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

Match the following List-$I$ with List-$II$ for $\frac{dy}{dx}$:

List-$I$	List-$II$
$A. x^2 + y^2 + 3xy = 7$	$I. \frac{x^2 + ay}{ax + y^2}$
$B. x^{2/3} + y^{2/3} = a^{2/3}$	$II. \frac{-(2x + 3y)}{3x + 2y}$
$C. x^3 + y^3 = 3axy$	$III. -(\frac{y}{x})^{1/3}$
$D. xy(x - y) = 2$	$IV. \frac{x^2 - ay}{ax - y^2}$
	$V. \frac{y(2x - y)}{x(x - 2y)}$

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