Limits and Deri...

Evaluate $$\displaystyle \lim_{n\rightarrow \infty }\left [ \left ( 1+\frac{1}{n^{2}} \right )\left ( 1+\frac{2^{2}}{n^{2}} \right )\left ( 1+\frac{3^{2}}{n^{2}} \right )......\left ( 1+\frac{n^{2}}{n^{2}} \right ) \right ]^{1/n}$$

If $$\displaystyle y=\left | \cos x \right |+\left | \sin x \right |$$ then $$\displaystyle \frac{dy}{dx}$$ at $$x=\dfrac{2\pi }{3}$$ is:

$$\displaystyle \frac{d}{d\theta }\left ( \tan ^{-1}\left ( \frac{1-\cos \theta }{\sin \theta } \right ) \right )$$ equals if $$\displaystyle-\pi <\theta <\pi $$

If $$y = \displaystyle \tan^{-1}\left (\cot x \right ) +\cot^{-1}(\tan x),$$ then $$\displaystyle \frac{dy}{dx}$$ is equal to-

$$\displaystyle \frac{d}{dx}\left ( \tan ^{-1}\left ( \frac{\sqrt{x}-x}{1+x^{3/2}} \right ) \right )$$ equals $$\displaystyle ($$for $$x\geq 0)$$

If $$y = sec  x^0$$ then $$\displaystyle \frac{dy}{dx} = $$

Let $$\displaystyle y=(1+x^{2})\tan^{-1}(x-x)$$ and $$\displaystyle f(x)=\frac1{2x}\frac {dy}{dx},$$ then $$f(x)+\cot^{-1}x$$ is equal to

$$\displaystyle \lim_{x \rightarrow \infty} (\sqrt{x^2 + 8x + 3} - \sqrt{x^2 + 4x + 3}) =$$

The limit of $$x\sin { \left( { e }^{ \frac { 1 }{ x }  } \right)  } $$ as $$x\rightarrow 0$$

If $$r=\left[2\phi +\cos^2\left(2\phi +\dfrac{\pi}4\right)\right]^{\tfrac12},$$ then what is the value of the derivative of $$\dfrac{dr}{d\phi}$$ at $$\phi=\dfrac{\pi}4?$$