$\frac{d}{dx}\left( \frac{\cot^2 x - 1}{\cot^2 x + 1} \right) = $

Differentiate $\left(x^{2}-5 x+8\right)\left(x^{3}+7 x+9\right)$ by expanding the product to obtain a single polynomial.

If $f(x)=x^{n}$,where $n$ is a non-negative integer,then the values of $n$ for which $f^{\prime}(\alpha+\beta)=f^{\prime}(\alpha)+f^{\prime}(\beta)$ for all $\alpha, \beta > 0$ is:

If $f:R \to R$ is a differentiable function and $f(1) = 4$,then the value of $\lim_{x \to 1} \int_4^{f(x)} \frac{2t}{x - 1} dt$ is:

Suppose $f(x)$ is differentiable at $x = 1$ and $\mathop {\lim }\limits_{h \to 0} \frac{1}{h}f(1 + h) = 5$,then $f'(1)$ equals

The differential coefficient of $f(\sin x)$ with respect to $x,$ where $f(x) = \log x,$ is