Answer
(a)
$$
S^{\prime}(1) =4
$$
(b)
$$
P^{\prime}(2) =6
$$
(c)
$$
Q^{\prime}(1) =\frac{7}{9}
$$
(d)
$$
C^{\prime}(2) =12
$$
Work Step by Step
(a) Since
$$
S(x)=f(x)+g(x)
$$
then
$$
\begin{aligned}
S^{\prime}(x)&=f^{\prime}(x)+g^{\prime}(x) \\
\Rightarrow S^{\prime}(1) &=f^{\prime}(1)+g^{\prime}(1) \\
& =3+1=4
\end{aligned}
$$
So,
$$
S^{\prime}(1) =4
$$
(b)
Since
$$
P(x)=f(x) g(x)
$$
then by using the Product Rule to differentiate the function $P$ we have :
$$
\begin{aligned}
P^{\prime}(x) &=f(x) g^{\prime}(x)+g(x) f^{\prime}(x) \\
\Rightarrow P^{\prime}(2) &=f(2) g^{\prime}(2)+g(2) f^{\prime}(2)\\
&=1(4)+1(2)\\
&=4+2=6 \\
\end{aligned}
$$
So,
$$
P^{\prime}(2) =6
$$
(c)
Since
$$
Q(x)=\frac{f(x)}{g(x)}
$$
then by using the Quotient Rule to differentiate the function $Q$ we have :
$$
\begin{aligned}
Q^{\prime}(x)&=\frac{g(x) f^{\prime}(x)-f(x) g^{\prime}(x)}{[g(x)]^{2}} \\
\Rightarrow Q^{\prime}(1) &=\frac{g(1) f^{\prime}(1)-f(1) g^{\prime}(1)}{[g(1)]^{2}} \\
& =\frac{3(3)-2(1)}{3^{2}}\\
&=\frac{9-2}{9}=\frac{7}{9}
\end{aligned}
$$
So,
$$
Q^{\prime}(1) =\frac{7}{9}
$$
(d)
Since
$$
C(x)=f(g(x))
$$
then by using the Chain Rule to differentiate the function $C$ we have :
$$
\begin{aligned}
C^{\prime}(x)&=f^{\prime}(g(x)) g^{\prime}(x) \\
\Rightarrow C^{\prime}(2)&=f^{\prime}(g(2)) g^{\prime}(2)\\
&=f^{\prime}(1) \cdot 4\\
&=3 \cdot 4=12
\end{aligned}
$$
So,
$$
C^{\prime}(2) =12
$$
