Part A. We are asked to determine the derivative of each of the functions. To do that we will use the chain rule:
[tex]\frac{d}{dx}f(g(x))=f^{\prime}(g(x))g^{\prime}(x)[/tex]
This means that we take the derivative of the function as a whole and then we multiply it by the derivative of the function inside the parenthesis.
Let's take Alice's function.
[tex]f(x)=15(4+3x)^4[/tex]
We have the following:
[tex]\begin{gathered} f(x)=15(g(x))^4 \\ g(x)=4+3x \end{gathered}[/tex]
Taking the derivative of f(x) we get:
[tex]f^{\prime}(x)=60(g(x))^3g^{\prime}(x)[/tex]
Now, we take the derivative of g(x):
[tex]g^{\prime}(x)=3[/tex]
Now, we substitute the values:
[tex]\begin{gathered} f^{\prime}(x)=60(4+3x)^3(3) \\ \\ f^{\prime}(x)=180(4+3x)^3 \end{gathered}[/tex]
And thus we get the derivative of the function.
The same procedure is done for the other functions.
Part B.
Let's take the function for Dani's guess:
[tex]f(x)=\frac{1}{18}(4+3x)^6[/tex]
Applying the chain rule we get:
[tex]f^{\prime}(x)=\frac{6}{18}(4+3x)^5(3)[/tex]
Simplifying we get:
[tex]\begin{gathered} f^{\prime}(x)=\frac{18}{18}(4+3x)^5 \\ \\ f^{\prime}(x)=(4+3x)^5 \end{gathered}[/tex]
Therefore, Dani was the student that was correct.
