Let $p$ and $q$ be piecewise linear functions given by their respective graphs below.
Let $r(x)=\frac{q(x)}{p(x)}$. Determine $r^{\prime}(0)$. Write your answer as an integer or

Step 1 ：Given that \(r(x)=\frac{q(x)}{p(x)}\), we can use the quotient rule to find the derivative. The quotient rule states that the derivative of \(\frac{u}{v}\) is \(\frac{vu'-uv'}{v^2}\), where \(u\) and \(v\) are functions of \(x\), and \(u'\) and \(v'\) are their respective derivatives.

Step 2 ：In this case, \(u=q(x)\), \(v=p(x)\), \(u'=q'(x)\), and \(v'=p'(x)\).

Step 3 ：So, \(r'(x)=\frac{p(x)q'(x)-q(x)p'(x)}{[p(x)]^2}\).

Step 4 ：To find \(r^{\prime}(0)\), we need to find \(p(0)\), \(q(0)\), \(p'(0)\), and \(q'(0)\).

Step 5 ：From the graphs, we can see that \(p(0)=1\) and \(q(0)=2\).

Step 6 ：The derivatives \(p'(x)\) and \(q'(x)\) represent the slopes of the functions \(p(x)\) and \(q(x)\) at a given point. From the graphs, we can see that the slopes of \(p(x)\) and \(q(x)\) at \(x=0\) are \(-1\) and \(3\) respectively. So, \(p'(0)=-1\) and \(q'(0)=3\).

Step 7 ：Substituting these values into the equation for \(r'(x)\), we get:

Step 8 ：\(r^{\prime}(0)=\frac{p(0)q'(0)-q(0)p'(0)}{[p(0)]^2}=\frac{1*3-2*-1}{1^2}=\frac{3+2}{1}=5\).

Step 9 ：So, \(r^{\prime}(0)=5\).

Step 10 ：The final answer is \(\boxed{5}\).