If $5 f(x) + 3 f\left(\frac{1}{x}\right) = x + 2$ and $y = x f(x)$,then $\frac{dy}{dx}$ at $x = 1$ is equal to

Find the derivative of the following function: $3 \cot x + 5 \operatorname{cosec} x$

Let $f$ and $g$ be differentiable functions on $R$ such that $f \circ g$ is the identity function. If for some $a, b \in R$, $g^{\prime}(a) = 5$ and $g(a) = b$, then $f^{\prime}(b)$ is equal to:

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

If $y = \tan^{-1}(\frac{1}{x^2+x+1}) + \tan^{-1}(\frac{1}{x^2+3x+3}) + \tan^{-1}(\frac{1}{x^2+5x+7}) + \dots$ upto $n$ terms, then $y'(0) = $

If $f(x) = \begin{cases} x-5, & \text{for } x \leq 1 \\ 4x^2-9, & \text{for } 1 < x < 2 \\ 3x+4, & \text{for } x \geq 2 \end{cases}$,then $f^{\prime}(2^{+})$ is equal to: