\begin{tabular}{|l|}\hline 2. Calcula el volumen \\ del sólido de \\ revolución que se \\ forma de girar sobre \\ el eje \( x \) la función \\ \( f(x)=x^{2}+x \) con \\ límites \( [-1,5] \)\end{tabular}

Find the derivative of the following function by first expanding the expression. \[ f(x)=(6 x+1)\left(5 x^{2}+7\right) \] \( f^{\prime}(x)=\square \)

Find the derivative of the following function. \( g(t)=4 \sqrt{t} \) \( g^{\prime}(t)=\square \)

Find an equation of the line tangent to the given curve at \( \mathrm{x}=\mathrm{a} \). \[ y=4+6 x+11 x e^{x} ; a=0 \] The equation for the tangent line is \( y=\square \).

Let \( f(x)=2 x^{3}+3 x^{2}-12 x+1 \). a. Find all points on the graph of \( f(x) \) at which the tangent line is horizontal. b. Find all points on the graph of \( f(x) \) at which the tangent line has slope 108 .

Find the intervals on which \( f \) is increasing and the intervals on which it is decreasing. \( f(x)=\frac{x^{4}}{4}-3 x^{3}+9 x^{2} \)