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

Let $P$ be a non-zero polynomial such that $P(1+x)=P(1-x)$ for all real $x$ and $P(1)=0$. Let $m$ be the largest integer such that $(x-1)^m$ divides $P(x)$ for all such $P(x)$. Then,$m$ equals

For some constants $a$ and $b$,find the derivative of $(x-a)(x-b)$.

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

If $y = \frac{ax+b}{cx+d}$ and $\frac{dx}{dy} = \frac{ad-bc}{Py^2+Qy+R}$,then $P+Q+R =$

Find the derivative of the following function (it is to be understood that $a, b, c, d, p, q, r$ and $s$ are fixed non-zero constants and $m$ and $n$ are integers): $\frac{a+b \sin x}{c+d \cos x}$