Question

Simplifique f(x+h)-f(x)/h, com h diferente de 0 sendo f(x) igual a f(x)=x^3

Answer 1

$\dfrac{f(x+h)-f(x)}{h}\\ \\ \\ =\dfrac{(x+h)^{3}-x^{3}}{h}$


Expandindo o binômio $(x+h)^{3},$

$=\dfrac{(x^{3}+3x^{2}h+3xh^{2}+h^{3})-x^{3}}{h}\\ \\ \\ =\dfrac{3x^{2}h+3xh^{2}+h^{3}}{h}\\ \\ \\ =\dfrac{\diagup\!\!\!\! h\cdot (3x^{2}+3xh+h^{2})}{\diagup\!\!\!\! h}\\ \\ \\ =3x^{2}+3xh+h^{2}.$


com $h \neq 0.$