If $y = (1 + x^2) \tan^{-1} x - x$,then $\frac{dy}{dx}$ is

If $y = \frac{5x}{\sqrt[3]{(1 - x)^2}} + \cos^2(2x + 1)$,then $\frac{dy}{dx} = $

If $y = \cot^{-1}(x^2)$,then $\frac{dy}{dx}$ is equal to

Let $f: R \rightarrow R$ be a differentiable function such that $|f(x) - f(y)| \leq 2|x - y|^{\frac{3}{2}}$ for all $x, y \in R$. If $f(0) = 1$,then $\int_0^1 f^2(x) dx = $

Let for a differentiable function $f:(0, \infty) \rightarrow \mathbb{R}$,$f(x)-f(y) \geq \log_e\left(\frac{x}{y}\right)+x-y, \forall x, y \in(0, \infty)$. Then $\sum_{n=1}^{20} f^{\prime}\left(\frac{1}{n^2}\right)$ is equal to

Differentiate the following with respect to $x$: $\frac{e^{x}}{\sin x}$