Question

solve 12 and label part A and B, make sure you do both a and b

Answer 1

ANSWER:

(a)

$g^{\prime}(s)=2s-2$

(b) The answer in part (b) agrees with that of part (a)

EXPLANATION:

Given:

$g(s)=\frac{4s^3-8s^2+4s}{4s}$

a) We'll go ahead and determine the derivative of g(s) as seen below;

$\begin{gathered} Let\text{ }u=4s^3-8s^2+4s \\ u^{\prime}(s)=12s^2-16s+4 \\ Let\text{ }v=4s \\ v^{\prime}(s)=4 \end{gathered}$

Let's go ahead and substitute the above values into the below Quotient Rule formula and simplify;

$\begin{gathered} g^{\prime}(s)=\frac{vu^{\prime}(s)-uv^{\prime}(s)}{u^2} \\ \\ =\frac{4s(12s^2-16s+4)-(4s^3-8s^2+4s)(4)}{(4s)^2} \\ \\ =\frac{48s^3-64s^2+16s-16s^3+32s^2-16s}{16s^2} \\ \\ =\frac{32s^3-32s^2}{16s^2} \\ \\ =\frac{32s^3}{16s^2}-\frac{32s^2}{16s^2} \\ \\ =2s-2 \\ \\ \therefore g^{\prime}(s)=2s-2 \end{gathered}$

b) Let's go ahead and simplify g(s) as seen below;

$\begin{gathered} g(s)=\frac{4s^{3}-8s^{2}+4s}{4s} \\ \\ =\frac{4s^3}{4s}-\frac{8s^2}{4s}+\frac{4s}{4s} \\ \\ \therefore g(s)=s^2-2s+1 \\ \\ g^{\prime}(s)=2s-2+0 \\ \\ \therefore g^{\prime}(s)=2s-2 \end{gathered}$

We can see from the above that the answer in part (b) agrees with that of part (a)