Application of ...

The curve $$y-{e}^{xy}+x=0$$ has a vertical tangent at the point

The slope of the curve $$y=\sin { x } +\cos ^{ 2 }{ x }$$ is zero at a point , whose x-coordinate can be ?

If the slope of one of the lines given by $${a^2}{x^2} + 2hxy+by^2 = 0$$ be three times of the other , then h is equal to 

If the curves $$\dfrac {x^{2}}{a^{2}} + \dfrac {y^{2}}{4} = 1$$ and $$y^{3} = 16x$$ intersect at right angles, then $$a^{2}$$ is equal to

If for a curve represented parametrically by $$x={ sec }^{ 2 }t,\quad y=cot\quad t\quad $$ , the tangent  at a point $$P(t=\frac { \pi  }{ 4 } )$$ meets the curve again at the point Q, then $$\begin{vmatrix} PQ \end{vmatrix}$$is equal to 

$$\dfrac { d } { d x } \left( \sin ^ { 5 } x \cdot \sin 5 x \right) =$$

Let f and g be non-increasing and non-decreasing functions respectively from $$[0,\infty ]$$ vto $$[0,\infty ]$$ and $$h(x)=f(g(x)),h(0)=0$$, then in $$[0,\infty ]$$, $$h(x)-h(1)$$ is 

The function $$f\left( x \right) = \frac{{\left| {x - 1} \right|}}{{{x^2}}}$$ is

$$y = 6\tan \,x\left( {{e^x} - x - 1} \right) - 3{x^3} - {x^4} - \frac{5}{4}{x^5},\,$$ if $${n^{th}}$$ derivative at x=0 is non zero then least value of n is

If $$(x+{ y }^{ 3 })\dfrac { dy }{ dx } $$=y and y(0)=2. then sum of all possible value(s) of y(1) is ________________.