If $f(1)=3$ and $f^{\prime}(1)=2$,then the value of $\frac{d}{d x}\left\{\log \left[f\left(e^x+2 x\right)\right]\right\}$ at $x=0$ is

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 . . . . . .

If $y^{\frac{1}{m}}+y^{-\frac{1}{m}}=2x$,where $x \neq 1$,then $(x^2-1)(\frac{dy}{dx})^2$ 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

The point which lies on the tangent drawn to the curve $x^4 e^y + 2 \sqrt{y+1} = 3$ at the point $(1,0)$ is

If $x=\sqrt{1-\tan y}$,then $\frac{dy}{dx}=$