A stone is thrown into a quite lake and the waves formed move in circles. If the radius of a circular wave increases at the rate of 4 cm/sec, then the rate of increase in its area, at the instant when its radius is 10 cm, is _________ cm$$^2$$/sec.

The function $$f(x)=\cot ^{-1} x+x$$ is increasing in the interval.

$$\lim _\limits{x \rightarrow 1} \frac{(2 x-3)(\sqrt{x}-1)}{2 x^2+x-3}=$$

The derivative of $$(\log x)^x$$ with respect to $$\log x$$ is