Continuity and ...

If $$y = {\left( {\sin \,x} \right)^x}$$, then $$\dfrac{{dy}}{{dx}} = $$

If $$ f(x) = \bigg[ \frac {a+x}{1+x} \bigg]^{a+1+2x} $$ then $$ {a^{a+1}} \bigg [ 2 \ log \ a + {\frac {1-a ^2}{a}} \bigg]$$ is

If a continuous function $$f$$ satisfies the relation 
$$\overset{t}{\underset{0}{\int}} \left(f(x) - \sqrt{f'(x)}\right)dx = 0$$ and $$f(0) = \dfrac{-1}{2}$$
Then $$f(x)$$ is equal to

Differentiate $${x^{\tan x}} + {{\mathop{\rm tanx}\nolimits} ^x}$$ with respect to $$x$$

If $$f:R \to R$$ be a differentiable function, such that $$f\left( {x + 2y} \right) = f\left( x \right) + f\left( {2y} \right) + 4xy$$ for all $$x,y \in R$$ then

Let $$f\left( x \right) = \left\{ \begin{array}{l}\begin{array}{*{20}{c}}{ - 1\,\,\, - }&{2\,\,\, \le \,\,\,{\rm{x}}}&\rangle &0\end{array}\\\begin{array}{*{20}{c}}{{x^2}\,\, - }&{1,\,0\,\,\,\,\rangle }&{{\rm{x}}\,\,\rangle }&2\end{array}\end{array} \right.$$ and g $$\left( x \right) = \left| {f\left( x \right)\left| { + f\left| {x\left. {} \right|} \right.} \right.} \right.$$ then the number of points which g(x) is non differentiable,is

If $$f(x)$$ is twice differentiable function such that $$f(a)=0, f(b)=2, f(c)=-1, f(d)=2, f(e)=0$$, where $$a<b<c<d<e$$, then the minimum number of zeroes of $$g(x)={(f'(x))}^{2}+f''(x).f(x)$$ in the interval $$[a, e]$$ is 

Let $$f\left( x \right) =\begin{cases} x\quad \quad \quad \quad \quad \quad \quad \quad x<1 \\ 2-x\quad \quad \quad \quad \quad \quad 1\le x\le 2 \\ -2+3x-{ x }^{ 2 }\quad \quad \quad x>2 \end{cases}$$ then $$f(x)$$ is

Let $$f(x)=\dfrac{1}{ax+b}$$ then $$f''(0)=$$