$\frac{d}{dx}(x^2 e^x \sin x) = $

Let $f: R \rightarrow R$ be defined by $f(x) = \log \left[e^x \left(\frac{x-2}{x+2}\right)^{3/4}\right]$. Find the value of $f'(0)$.

If $y = \frac{e^x \log x}{x^2}$,then $\frac{dy}{dx} =$

If $y=x^x+x^7+7^x+7^7$,then $\frac{dy}{dx}=$

If $y=(1+x)(1+x^2)(1+x^4) \dots (1+x^{2^n})$, then $\left(\frac{dy}{dx}\right)_{x=0}$ is equal to

If $f: R \rightarrow R$ is defined by $f(x) = \begin{cases} \frac{x-2}{x^2-3x+2} & \text{if } x \in R - \{1, 2\} \\ 2 & \text{if } x = 1 \\ 1 & \text{if } x = 2 \end{cases}$,then find $\lim_{x \rightarrow 2} \frac{f(x)-f(2)}{x-2}$.