.
.
.![\begin{aligned} \frac{d^3}{{d x}^3}\left[22\, x^3 + 28\, x^2 + 55\, x + 70\right]\end{aligned}](/tpl/images/0869/9166/5525c.png)
.![\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}](/tpl/images/0869/9166/1d96c.png)
.![\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}](/tpl/images/0869/9166/54670.png)
.
denote an integer. By the power rule:![\displaystyle \frac{d}{d x} \left[ x^{n}\right] = n\, x^{n-1}](/tpl/images/0869/9166/9b5b4.png)
.![\displaystyle \frac{d^3}{d x^3}\, \left[ x^3 \right]](/tpl/images/0869/9166/c6a15.png)
, ![\displaystyle \frac{d^3}{d x^3}\, \left[ x^2 \right]](/tpl/images/0869/9166/c1e35.png)
, and ![\displaystyle \frac{d^3}{d x^3}\, \left[ x \right]](/tpl/images/0869/9166/e2044.png)
.![\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](/tpl/images/0869/9166/42dd6.png)
.![\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](/tpl/images/0869/9166/57d33.png)
(the first derivative of a constant is zero.)![\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](/tpl/images/0869/9166/1f62b.png)
.![\displaystyle \frac{d^3}{{d x}^3} \left[70\right] = 0](/tpl/images/0869/9166/5658f.png)
.
.![\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}](/tpl/images/0869/9166/7c641.png)
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