A ladder 5 m long rests against a vertical wall. If its top slides downwards at the rate of $10 \mathrm{~cm} / \mathrm{sec}$., then the foot of the ladder is sliding at the rate of _________ $\mathrm{m} / \mathrm{sec}$., when it is 4 m away from the wall.

The equation of the tangent to the curve $y=1-\mathrm{e}^{\frac{x}{3}}$ at the point of intersection with Y -axis is

If $8 \mathrm{f}(x)+6 \mathrm{f}\left(\frac{1}{x}\right)=x+5$ and $y=x^2 \mathrm{f}(x)$, then $\frac{\mathrm{d} y}{\mathrm{~d} x}$ at $x=-1$ is

The function $f(x)=\frac{\log _e(\pi+x)}{\log _e(e+x)}$ is