$\frac{d}{d x}\left(e^{\log _e \sqrt{1+\tan ^2 x}}\right) =$

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

Find the derivative of the function: $\frac{x^{2} \cos \left(\frac{\pi}{4}\right)}{\sin x}$

Differentiate the function with respect to $x$: $\sin^{-1}(x\sqrt{x})$,where $0 \le x \le 1$.

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): $(ax + b)^n (cx + d)^m$

If $f$ is differentiable, $f(x+y)=f(x) f(y)$ for all $x, y \in R$, $f(3)=3$, and $f^{\prime}(0)=11$, then $f^{\prime}(3)$ is equal to: