The derivative of a function f(x) is defined as the limit,
[tex]f'(x):=\displaystyle\lim_{h\to0}\frac{f(x+h)-f(x)}h[/tex]
With f(x) = x⁶, we have
[tex]f'(x)=\displaystyle\lim_{h\to0}\frac{(x+h)^6-x^6}h[/tex]
Expand f(x + h) in the numerator:
(x + h)⁶ = x⁶ + 6x⁵h + 15x⁴h² + 20x³h³ + 15x²h⁴ + 6xh⁵ + h⁶
so that the x⁶ terms cancel, leaving us with
[tex]f'(x)=\displaystyle\lim_{h\to0}\frac{6x^5h+15x^4h^2+20x^3h^3+15x^2h^4+6xh^5+h^6}h[/tex]
h is approaching 0, so h ≠ 0 and we can cancel the common factor in the numerator and denominator:
[tex]f'(x)=\displaystyle\lim_{h\to0}\left(6x^5+15x^4h+20x^3h^2+15x^2h^3+6xh^4+h^5\right)[/tex]
Now as h converges to 0, each term containing h vanishes, leaving us with
f'(x) = 6x⁵
