If g(x) = 1/(3x²), the by the definition of the derivative, we have
[tex]\displaystyle g'(x) = \lim_{h\to0} \frac{g(x+h) - g(x)}h[/tex]
[tex]\displaystyle g'(x) = \lim_{h\to0} \frac{\frac1{3(x+h)^2} - \frac1{3x^2}}h[/tex]
[tex]\displaystyle g'(x) = \lim_{h\to0} \frac{\frac{x^2}{3x^2(x+h)^2} - \frac{(x+h)^2}{3x^2(x+h)^2}}h[/tex]
[tex]\displaystyle g'(x) = \lim_{h\to0} \frac{x^2 - (x+h)^2}{3x^2(x+h)^2h}[/tex]
[tex]\displaystyle g'(x) = \lim_{h\to0} \frac{x^2 - x^2 - 2xh - h^2}{3x^2(x+h)^2h}[/tex]
[tex]\displaystyle g'(x) = - \lim_{h\to0} \frac{2xh + h^2}{3x^2(x+h)^2h}[/tex]
[tex]\displaystyle g'(x) = - \lim_{h\to0} \frac{2x + h}{3x^2(x+h)^2}[/tex]
[tex]\displaystyle g'(x) = - \frac{2x}{3x^4} = \boxed{-\frac2{3x^3}}[/tex]
