If $y = \tan^{-1} \sqrt{\frac{1 + \sin 2x}{1 - \sin 2x}}$, then at $x = \frac{\pi}{6}$, the value of $\frac{dy}{dx}$ is

For the function $f(x) = \frac{x^{100}}{100} + \frac{x^{99}}{99} + \dots + \frac{x^2}{2} + x + 1$,prove that $f^{\prime}(1) = 100 f^{\prime}(0)$.

If $y = x^2 + \cos(2x) + e^{ax}$,then find $\frac{dy}{dx}$.

If $f(x)$ is an odd differentiable function defined on $(-\infty, \infty)$ such that $f^{\prime}(3)=2,$ then $f^{\prime}(-3)$ is equal to

Find the derivative of the following function (it is to be understood that $a, b, c, d, p, q, r$ and $s$ are fixed non-zero constants and $m$ and $n$ are integers): $\frac{\sin (x+a)}{\cos x}$

Suppose a differentiable function $f(x)$ satisfies the identity $f(x+y) = f(x) + f(y) + xy^2 + x^2y$ for all real $x$ and $y$. If $\lim_{x \rightarrow 0} \frac{f(x)}{x} = 1$,then $f'(3)$ is equal to: