Answer:
[tex]f^{\prime\prime\prime}(x) = 132[/tex].
Step-by-step explanation:
Start by expanding the expression of [tex]f(x)[/tex].
[tex]\begin{aligned} f(x) &= (2\, x^2 + 5) \, (11\, x+ 14) \\ &= 2\, x^2\, (11\, x + 14) + 5\, (11\, x + 14)\\&= \left(22\, x^3 + 28\, x^2\right) + \left(55\, x + 70\right) = 22\, x^3 + 28\, x^2 + 55\, x + 70\end{aligned}[/tex].
The question is asking for the third derivative of that expression:
[tex]\begin{aligned} \frac{d^3}{{d x}^3}\left[22\, x^3 + 28\, x^2 + 55\, x + 70\right]\end{aligned}[/tex].
That's equivalent to finding:
[tex]\begin{aligned} \frac{d^3}{{d x}^3}\left[22\, x^3\right] + \frac{d^3}{{d x}^3} \left[28\, x^2\right] + \frac{d^3}{{d x}^3} \left[55\, x\right] + \frac{d^3}{{d x}^3} \left[70\right]\end{aligned}[/tex].
Move the constants outside of the derivative operator:
[tex]\begin{aligned} 22\, \left(\frac{d^3}{{d x}^3}\left[x^3\right]\right) + 28\, \left(\frac{d^3}{{d x}^3} \left[x^2\right]\right) + 55\, \left(\frac{d^3}{{d x}^3} \left[x\right]\right) + \frac{d^3}{{d x}^3} \left[70\right]\end{aligned}[/tex].
Let [tex]n[/tex] denote an integer. By the power rule:
[tex]\displaystyle \frac{d}{d x} \left[ x^{n}\right] = n\, x^{n-1}[/tex].
Apply these two rules repeatedly to find [tex]\displaystyle \frac{d^3}{d x^3}\, \left[ x^3 \right][/tex], [tex]\displaystyle \frac{d^3}{d x^3}\, \left[ x^2 \right][/tex], and [tex]\displaystyle \frac{d^3}{d x^3}\, \left[ x \right][/tex].
[tex]\displaystyle \frac{d^3}{{d x}^3}\, \left[ x^3 \right] = \frac{d^2}{{d x}^2} \left[3\, x^2\right] = \frac{d}{{d x}} \left[6\, x\right] = 6[/tex].
[tex]\displaystyle \frac{d^3}{{d x}^3}\, \left[ x^2 \right] = \frac{d^2}{{d x}^2} \left[2\, x\right] = \frac{d}{{d x}} \left[ 1 \right] = 0[/tex] (the first derivative of a constant is zero.)
[tex]\displaystyle \frac{d^3}{{d x}^3}\, \left[ x \right] = \frac{d^2}{{d x}^2} \left[1\right] = \frac{d}{{d x}} \left[0 \right] = 0[/tex].
Similarly, [tex]\displaystyle \frac{d^3}{{d x}^3} \left[70\right] = 0[/tex].
Substitute these back to the expression of [tex]f^{\prime\prime\prime}(x)[/tex].
[tex]\begin{aligned} f^{\prime\prime\prime} &= 22\, \underbrace{\left(\frac{d^3}{{d x}^3}\left[x^3\right]\right)}_{6} + \underbrace{28\, \left(\frac{d^3}{{d x}^3} \left[x^2\right]\right)}_0 + \underbrace{55\, \left(\frac{d^3}{{d x}^3} \left[x\right]\right)}_0 + \underbrace{\frac{d^3}{{d x}^3} \left[70\right]}_0 \\ &= 22 \times 6 \\ &= 132\end{aligned}[/tex].
