If $f(x) = e^{x} g(x)$,$g(0) = 2$,and $g^{\prime}(0) = 1$,then $f^{\prime}(0)$ is

The left-hand derivative of $f(x) = \{x\} \sin(\pi x)$ at $x = k$ ($k$ is an integer) is (where $\{ \}$ denotes the fractional part function).

The differential coefficient of $\sec^{-1} x$ is:

The differential coefficient of $\log _{10} x$ with respect to $\log _{x} 10$ is

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

Find the derivative: $\frac{d}{dx}[\cos((1 - x^2)^2)] = ?$