Application of ...

If the function $$f(x)=2\tan x+(2a+1)\log _{ e }{ \left| \sec { x }  \right|  } +(a-2)x$$ is increasing on $$R$$, then

Consider the equation $$x^y=e^{x-y}$$
What is $$\dfrac{d^2y}{dx^2}$$ at $$x=1$$ equal to ?

Consider the equation $$x^y=e^{x-y}$$
What is $$\dfrac{dy}{dx}$$ at $$x=1$$ equal to ?

A curve $$y=me^{mx}$$ where $$m > 0$$ intersects y-axis at a point $$P$$.
What is the slope of the curve at the point of intersection $$P$$ ? 

A curve $$y=me^{mx}$$ where $$m > 0$$ intersects y-axis at a point $$P$$.
How much angle does the tangent at $$P$$ make with y-axis ? 

The function $$f(x)=4-3x+3x^2-x^3$$ is

The real value of $$k$$ for which $$f(x)=x^2+kx+1$$ is increasing on $$(1, 2)$$, is 

$$f(x)=\sin x-kx $$ is decreasing for all $$x \in R$$, when

The function $$f(x)=3x+\cos 3x$$ is

The slope of the tangent to the curve $$y = \sqrt{4-x^{2}}$$ at the point, where the ordinate and the abscissa are equal , is