If $f(x)$ is an even function and $f^{\prime}(x)$ exists,then $f^{\prime}(e) + f^{\prime}(-e)$ is

If $y(\alpha)=\sqrt{2\left(\frac{\tan \alpha+\cot \alpha}{1+\tan ^{2} \alpha}\right)+\frac{1}{\sin ^{2} \alpha}}$ for $\alpha \in\left(\frac{3 \pi}{4}, \pi\right)$,then find $\frac{d y}{d \alpha}$ at $\alpha=\frac{5 \pi}{6}$.

If $f: R \rightarrow R$ is an even function having derivatives of all orders, then which of the following is an odd function?

If $e^{x}=y+\sqrt{y^2-1}$, then $\frac{d y}{d x}=$

Let $f: R \rightarrow R$ satisfy the equation $f(x+y)=f(x) \cdot f(y)$ for all $x, y \in R$ and $f(x) \neq 0$ for any $x \in R$. If the function $f$ is differentiable at $x=0$ and $f'(0)=3$,then $\lim_{h \rightarrow 0} \frac{1}{h}(f(h)-1)$ is equal to ....... .

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