Answer
a) $F'(2) = -\frac{3}{5}$
b) The red is the graph of $ f(x) = \frac{5x}{1+x^2}$
The blue is the tangent line $ y = -\frac{3x}{5} + \frac{16}{5}$.
Work Step by Step
$F'(a) = \lim\limits_{x \to a} \frac{f(x)-f(a)}{x-a}$
$F'(a) = \lim\limits_{x \to a} \frac{\frac{5x}{1+x^2}-\frac{5a}{1+a^2}}{(x-a)}$
$F'(a) = \lim\limits_{x \to a} \frac{\frac{5x}{1+x^2}-\frac{5a}{1+a^2}}{(x-a)}$ $\times {\frac{(1+x^2)(1+a^2)}{(1+x^2)(1+a^2}}$
$F'(a) = \lim\limits_{x \to a} \frac{5x(1+a^2)-5a(1+x^2)}{(x-a)(1+x^2)(1+a^2)}$
$F'(a) = \lim\limits_{x \to a} \frac{5x+5xa^2-5a-5x^2)}{(x-a)(1+x^2)(1+a^2)}$
Simplify the equation:
$F'(a) = \lim\limits_{x \to a} \frac{5[(xa^2-a)-(x^2a-x)]}{(x-a)(1+x^2)(1+a^2)}$
$F'(a) = \lim\limits_{x \to a} \frac{5[(a(xa-1))-x(xa-1)]}{(x-a)(1+x^2)(1+a^2)}$
$F'(a) = \lim\limits_{x \to a} \frac{5[(a-x)(xa-1)]}{(x-a)(1+x^2)(1+a^2)}$
Multiply $ -1$ to $(a-x)$ to make it $(x-a)$
$F'(a) = \lim\limits_{x \to a} \frac{-5[(a-x)(xa-1)]}{(x-a)(1+x^2)(1+a^2)}$
$F'(a) = \lim\limits_{x \to a} \frac{-5[(x-a)(xa-1)]}{(x-a)(1+x^2)(1+a^2)}$
$F'(a) = \lim\limits_{x \to a} \frac{-5[(xa-1)]}{(1+x^2)(1+a^2)}$
Substitute $x$ for $a$
$F'(a) = \lim\limits_{x \to a} \frac{-5(a*a-1)}{(1+a^2)(1+a^2)}$
$F'(a) = \frac{-5(a^2-1)}{(1+a^2)^2}$
$F'(2) = \frac{-5(2^2-1)}{(1+2^2)^2}$
$F'(2) = \frac{-5(4-1)}{(1+4)^2}$
$F'(2) = \frac{-5(3)}{5^2}$
$F'(2) = -\frac{-15}{25}$
$F'(2) = -\frac{3}{5}$
Tangent line: $ y - y_{1} = m(x-x_{1})$
$ y - 2 = -\frac{3}{5}(x-2)$
$ y = -\frac{3}{5}(x-2) + 2$
$ y = -\frac{3}{5}x + \frac{6}{5}+2$
$ y = -\frac{3}{5}x + \frac{16}{5}$
