If ${x^2} + {y^2} = 1$,then find the relation between $y'$ and $y''$,where $y' = \frac{dy}{dx}$ and $y'' = \frac{d^2y}{dx^2}$.

If $\log (x+y)=\log (x y)+a$,where $a$ is a constant,then $\frac{dy}{dx}$ at $x=2$ and $y=4$ is

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

Let $f(x), x \in [0, \infty)$ be a non-negative continuous function. If $f'(x) \cos x \le f(x) \sin x$ for all $x \ge 0$,then the value of $f(2\pi)$ is equal to

If $ x^{y}=e^{x-y} $ then $ \frac{d y}{d x} $ is equal to

Let $f : (0, \infty) \rightarrow \mathbb{R}$ be a twice differentiable function. If for some $a \neq 0$,$\int_0^1 f(\lambda x) d\lambda = a f(x)$,$f(1) = 1$ and $f(16) = \frac{1}{8}$,then $16 - f^{\prime}\left(\frac{1}{16}\right)$ is equal to . . . . . .