Differentiation...

Differentiate $$\displaystyle x^{\sin^{-1}x}$$ w.r.t. $$\displaystyle \sin ^{-1}x.$$

If $$y = \displaystyle (\tan x)^{\log x}$$, then $$\cfrac{dy}{dx} = $$

If $$2f\left( \sin { x }  \right) +f\left( \cos { x }  \right) =x$$, then $$\displaystyle \frac { d }{ dx } f\left( x \right)$$ is

Differentiate $$\displaystyle \tan x^{n}+\tan ^{n}x-\tan ^{-1}\frac{a+x^{n}}{1-ax^{n}}.$$

A curve passing through the point $$(1,1)$$ is such that the intercept made by a tangent to it on x-axis is three times the x co-ordinate of the point of tangency, then the equation of the curve is:

Differentiate the following: $$\displaystyle \cot ^{-1}\frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{\left ( 1+\sin x \right )}-\sqrt{\left ( 1-\sin x \right )}}$$

Let $$ \displaystyle f\left ( x \right ) $$ be defined by $$ \displaystyle f\left ( x \right )=\left\{\begin{matrix}\sin 2x & \text{if } 0< x\leq \dfrac{\pi}6\\ ax+b& \text{if } \dfrac{\pi}6< x\leq 1\end{matrix}\right. $$. The values of $$a$$ and $$b$$ such that $$ \displaystyle f $$ and $$ \displaystyle {f}' $$ are continuous, are

A polynomial $$f(x)$$ leaves remainder $$15$$ when divided by $$(x-3)$$ and $$(2x+1)$$ when divided by $$(x-1)^2$$. When $$f$$ is divided by $$(x-3)(x-1)^2,$$ the remainder is

If $$f\left( x \right) $$ is a polynomial of degree $$n(>2)$$ and $$f\left( x \right) =f\left( k-x \right) ,($$ where $$k$$ is a fixed real number$$),$$ then degree of $$f'(x)$$ is

If for all $$x,y$$ the function $$f$$ is defined by $$f\left( x \right)+f\left( y \right)+f\left( x \right).f\left( y \right)=1$$ and $$f\left( x \right)>0$$, then