Solution
Step 1
Write the function
[tex]f(x)=\int_0^x\left(t^3+4t^2+6\right)dt\:[/tex]
Step 2
[tex]\begin{gathered} \mathrm{Apply\:the\:Sum\:Rule}: \\ \int f\left(x\right)\pm g\left(x\right)dx=\int f\left(x\right)dx\pm \int g\left(x\right)dx \\ \\ (\frac{t^4}{4}+\frac{4t^3}{3}+6t)_0^x \\ \\ =\frac{x^4}{4}+\frac{4x^3}{3}+6x \end{gathered}[/tex]
Step 3
[tex]\begin{gathered} \\ f^{\prime}(x)=\frac{d}{dx}\left(\frac{x^4}{4}+\frac{4x^3}{3}+6x\right) \\ \\ f^{\prime}(x)=x^3+4x^2+6 \\ \\ f^{\prime}^{\prime}(x)=\frac{d}{dx}\left(x^3+4x^2+6\right) \\ \\ f^{\prime}^{\prime}(x)=3x^2+8x \end{gathered}[/tex]
Final answer
[tex]f^{\prime}^{\prime}(x)=3x^2+8x[/tex]
