Find the third derivative of $f(x)=\sqrt[3]{12 x+1}$

Step 1 ：First, we need to rewrite the function $f(x)=\sqrt[3]{12x+1}$ in a form that is easier to differentiate. We can write it as $f(x)=(12x+1)^{\frac{1}{3}}$

Step 2 ：Next, we find the first derivative of $f(x)$ using the chain rule. The chain rule states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function. So, $f'(x)=\frac{1}{3}(12x+1)^{-\frac{2}{3}} \cdot 12 = 4(12x+1)^{-\frac{2}{3}}$

Step 3 ：Then, we find the second derivative of $f(x)$, again using the chain rule. $f''(x)=-\frac{8}{3}(12x+1)^{-\frac{5}{3}} \cdot 12 = -32(12x+1)^{-\frac{5}{3}}$

Step 4 ：Finally, we find the third derivative of $f(x)$, once more using the chain rule. $f'''(x)=\frac{160}{3}(12x+1)^{-\frac{8}{3}} \cdot 12 = \boxed{640(12x+1)^{-\frac{8}{3}}}$