Limits and Deri...

$$y=sin(x^2): \dfrac{dy}{dx}=$$

The solution of $$\cos y +(x \sin y-1)\dfrac{dy}{dx}=0$$ is 

If $$\mathop {\lim }\limits_{x \to \infty } \left( {\frac{{{x^2} + x + 1}}{{x + 1}} - ax - b} \right)\, = 4$$,then

Find derivative of given function w.r.t. the respective independent variable $$y= \frac{(sinx+cosx)}{cosx}$$

$$If {A_i} = \frac{{x - {a_i}}}{{\left| {x - {a_i}} \right|}}, \,i = 1,2,3,.....n$$ and $${a_1}< {a_2}< {a_3}....< {a_{n,}} \, then$$
$$\mathop {\lim }\limits_{x \to {a_m}} \left( {{A_1}{A_2}......{A_n}} \right), 1 \le m \le n$$

$$\displaystyle \mathop {\lim }\limits_{x \to \frac{\pi }{2}} \frac{{\cot x - \cos x}}{{{{\left( {\frac{\pi }{2} - x} \right)}^3}}}$$

Let p= $$\lim_{x\rightarrow 0+}(1+tan^{2}\sqrt{x})^{\frac{1}{2x}}$$ then log p is equal to :

Find that $$\frac { d } { d x } \left[ \frac { 2 } { \pi } \sin x ^ { 0 } \right] = ?$$

Consider $$A=\begin{bmatrix} \cos { \theta  }  & \sin { \theta  }  \\ -\sin { \theta  }  & \cos { \theta  }  \end{bmatrix}$$, then the value of $$\lim_{n \rightarrow \infty} \dfrac{A^{n}}{n}$$ (where $$\theta \in R$$) is equal to 

$$\dfrac{d}{dx}\left\{\tan^{-1}\dfrac{\sqrt{x}+\sqrt{a}}{1-\sqrt{ax}}\right\}=$$