Math. Please help with simplifying

Question

If f(x) = x - 6x^2 and h ≠ 0 how would you simply it as f(x + h) and f(x+ h) -f(x)/ h. Thank you in advance.

Answer 1

f(x+h) = (x+h) - 6(x+h)^2 = x+h -6(x^2 +2xh + h^2) = x+h-6x^2 +12xh +6h^2

f(x+h)-f(x) = x+h-6x^2 -12xh-6h^2 - (x-6x^2) =

x+h-6x^2 -12xh -6h^2 - x +6x^2 =

h-12xh-6h^2

[f(x+h)-f(x)]/h = (h-12xh-6h^2)/h = 1-12x -6h

limit of [f(x+h)-f(x)]/h as h approaches zero is 1-12x - 6(0) = 1-12x

(this is all to find the derivative of f(x) = f'(x) = 1-12x

If f(x) =x-6x^2, the short cut rule is take the exponent and use it as a coefficient and reduce the exponent by 1: x^1 - 6x^2 becomes 1x^0 - 2(6)x^1 = 1-12x)