sketch graph of derivative function from original function let us consider following  question  given on  image

because slope  is  equal to  rise over run, then we can calculate  slope at point  $x=-3$ by takeing following points
$(-3,0),(-1,2) so we have following form
$(2-0)/(-1-(-3))=2/2=1 $
is  part one correct?  and sketching of graph of derivatives will be simple plotting of values right?is there  more simple approach to do same? can i  estimate function itself?maybe i can guess that it is sinusoidal or some polynomial function and then  by inserting points  estimate its coefficients? thanks in advance

From the graph the function $f$ seems an odd function, so its derivative $f'$ is even. This means that:
$$
f'(-3)=f'(3) \qquad f'(-2)=f'(2) \qquad f'(-1)=f'(1)=0
$$
we can also see that $f'(x)<0$ for $-1<x<1$ and it has a minimum for $x=0$ and it seems positive for the values $x<-1$ and $x>1$ ( but the graph is limited so we can not be sure about this).
From all this we can assume that (in the simpler case) the derivative has the form $f'(x)=k(x^2-1)$ for $k>0$ and the function has the form $f(x)=\frac{k}{3}x(x^3-3)$
