If $f(x) = \sin([\pi^{2}]x) + \cos([-\pi^{2}]x)$,then find $f'(x)$,where $[\cdot]$ denotes the greatest integer function.

Suppose a differentiable function $f(x)$ satisfies the identity $f(x+y) = f(x) + f(y) + xy^2 + x^2y$ for all real $x$ and $y$. If $\lim_{x \rightarrow 0} \frac{f(x)}{x} = 1$,then $f'(3)$ is equal to:

If $y = \sin (\sqrt {\sin x + \cos x} )$,then $\frac{dy}{dx} = $

The differential coefficient of ${x^6}$ with respect to ${x^3}$ is

Let $f(x) = x - \frac{1}{x}$,then $f^{\prime}(-1)$ 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).