Let $f(x+y)=f(x) \cdot f(y)$ for all $x, y \in R$. Suppose that $f(3)=3$ and $f^{\prime}(0)=11$,then $f^{\prime}(3)$ is given by

$A$ real-valued function $f(x)$ satisfies the functional equation $f(x - y) = f(x)f(y) - f(a - x)f(a + y)$,where $a$ is a given constant and $f(0) = 1$. Then $f(2a - x)$ is equal to:

If $f(x)$ satisfies the relation $f\left( \frac{5x - 3y}{2} \right) = \frac{5f(x) - 3f(y)}{2}$ for all $x, y \in R$,with $f(0) = 1$ and $f'(0) = 2$,then the period of $\sin(f(x))$ is:

Let $f$ be a function defined by $f(xy) = \frac{f(x)}{y}$ for all positive real numbers $x$ and $y$. If $f(30) = 20$,then $f(40) = $

Let $f: N \rightarrow N$ be a function such that $f(x+y)=f(x)+f(y)+xy$ for every $x, y \in N$. If $f(1)=2$,then $\sum_{k=1}^{10} f(k)=$

Let $f(x)$ be a differentiable function which satisfies the equation $f(xy) = f(x) + f(y)$ for all $x > 0, y > 0$. Then $f'(x)$ is equal to: