mathematics

Calculating a Derivative by Definition

Suppose \$ f(x) = ax^2 + bx + c \$ (you can set \$ a=1, b=0, c=0 \$ for \$ f(x) = x^2 \$, but we’ll keep it general). $$ f(x + h) = a(x + h)^2 + b(x + h) + c = a(x^2 + 2xh + h^2) + b(x + h) + c $$ $$ = ax^2 + 2axh + ah^2 + bx + bh + c $$ $$ f(x) = ax^2 + bx + c $$