If $\sin ^2 x + \cos ^2 y = 1$,then $\frac{dy}{dx} = $

If $y = (1 + \frac{1}{x}) (1 + \frac{2}{x}) (1 + \frac{3}{x}) . . . . . . (1 + \frac{n}{x})$ and $x \neq 0$. When $x = -1$,find $\frac{dy}{dx}$.

If $f(x) = \frac{1}{x^2} \int_3^x (2t - 3f'(t)) dt$, then $f'(3)$ is equal to

If $\frac{y}{x} \cos^4 \alpha + \frac{x}{y} \sin^4 \alpha = 2 \sin^2 \alpha \cos^2 \alpha$,then $\frac{dy}{dx} = $

If $\sin (xy) + \frac{x}{y} = {x^2} - y,$ 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