Continuity and ...

The derivative of $$cosec^{-1} \left( \dfrac{1}{2x^2 - 1} \right )$$ with respect to $$\sqrt{1 - x^2}$$ at $$x = \dfrac{1}{2}$$ is

$$x$$	$$1$$	$$2$$	$$3$$	$$4$$	$$5$$
$$f(x)$$	$$4$$	$$3$$	$$7$$	$$1$$	$$3$$

The function f is continuous on the closed interval $$[1, 5]$$ and values of the function are shown in the table above. If the values in the table are used to calculate a trapezoidal sum, the approximate value of $$\int_{1}^{5}f(x)dx$$ is

The derivative of $$\displaystyle (\tan x)^{x}$$ is equal to-

If $$f(x) = \left\{\begin{matrix}\dfrac {1 - \sin x}{(\pi - 2x)^{2}} \cdot \dfrac {\log \sin x}{\log (1 + \pi^{2} - 4\pi x + 4x^{2})},& x\neq \dfrac {\pi}{2}\\ k, & x = \dfrac {\pi}{2}\end{matrix}\right.$$ is continuous at $$x = \dfrac {\pi}{2}$$, then $$k$$ is equal to

Find derivative of $$\tan^{-1}\dfrac{\cos x-\sin x}{\cos x+\sin x}$$ w.r.t. $$x$$.

If $$y=\sqrt { \left( a-x \right) \left( x-b \right)  } -\left( a-b \right) \tan ^{ -1 }{ \sqrt { \dfrac { a-x }{ x-b }  }  } $$, then $$\dfrac { dy }{ dx } $$ is equal to

Let $$f(x) = \left\{\begin{matrix}2a - x, &if\ -a < x < a \\ 3x - 2a, &if\ a \leq x \end{matrix}\right.$$. Then, which of the following is true?

If $$x = A\cos 4t + B\sin 4t$$, then $$\dfrac {d^{2}x}{dt^{2}}$$ is equal to

The function $$f(x)=\left[ { x }^{ 2 } \right] +{ \left[ -x \right]  }^{ 2 }$$, where $$[]$$ denotes the greatest integer function is

Find the derivative of $$\dfrac{\tan^{-1}x}{1+\tan^{-1}x}$$ w.r.t. $$\tan^{-1}x$$.