$f: R \rightarrow R$ is a function such that $f(0)=1$ and for all $x, y \in R$,$f(xy+1)=f(x)f(y)-f(y)-x+2$. Then $\frac{df}{dx}$ at $x=e$ is:

Let $f$ be a non-zero real-valued continuous function satisfying $f(x+y) = f(x) \cdot f(y)$ for all $x, y \in R$. If $f(2) = 9$,then $f(6)$ is equal to

Let $R$ denote the set of all real numbers. Let $a_i, b_i \in R$ for $i \in \{1, 2, 3\}$. Define the functions $f: R \rightarrow R$,$g: R \rightarrow R$,and $h: R \rightarrow R$ by $f(x) = a_1 + 10x + a_2x^2 + a_3x^3 + x^4$ and $g(x) = b_1 + 3x + b_2x^2 + b_3x^3 + x^4$. Let $h(x) = f(x+1) - g(x+2)$. If $f(x) \neq g(x)$ for every $x \in R$,then the coefficient of $x^3$ in $h(x)$ is:

Find $\sum_{t=1}^{39} f(t)$ if $f: R \rightarrow R$ is defined as $f(x+y)=f(x)+f(y)$ for all $x, y \in R$ and $f(1)=7$.

If $f(x+2y, x-2y) = xy$,then $f(x, y)$ is equal to

If $f(x)$ is a function satisfying the condition $f(x) = \frac{1}{3}\left[ f(x + 6) + \frac{6}{f(x + 7)} \right]$ and $f(x) \geq 0$ for all $x \in R$. If $\lim_{x \to \infty} f(x) = \sqrt{m}$,then the value of $m$ is: