For this problem, we are given a function and we need to determine the derivative on the given point.
The function is:
[tex]f(x)=(x^2+9)(6x+2)[/tex]
The derivative is:
[tex]\begin{gathered} f^{\prime}(x)=(x^2+9)^{\prime}(6x+2)+(x^2+9)(6x+2)^{\prime}\\ \\ f^{\prime}(x)=2x(6x+2)+(x^2+9)(6)\\ \\ f^{\prime}(x)=12x^2+4x+6x^2+54\\ \\ f^{\prime}(x)=18x^2+4x+54 \end{gathered}[/tex]
Now we need to apply the given point, which means replacing "x" with 2 and evaluating the expression.
[tex]\begin{gathered} f^{\prime}(2)=18(2)^2+4(2)+54\\ \\ f^{\prime}(2)=18\cdot4+8+54\\ \\ f^{\prime}(2)=72+62=134 \\ \end{gathered}[/tex]
For this problem we used the following differentiation rules: power rule and product rule.
