Question

Utilizando la definición de derivada(límites), Determine:
F(x)= C/ X^2

Answer 1

Respuesta:

Primero observa que

$\dfrac{\dfrac{c}{(x+h)^2}-\dfrac{c}{x^2}}{h}=\dfrac{\dfrac{x^2c-c(x+h)^2}{x^2(x+h)^2}}{h}=\dfrac{x^2c-c(x+h)^2}{hx^2(x+h)^2}=\dfrac{x^2c-c(x^2+2xh+h^2)}{hx^2(x^2+2xh+h^2)}=\\\\\dfrac{x^2c-cx^2-2cxh-ch^2}{hx^2(x^2+2xh+h^2)}=\dfrac{hc(-2x-h)}{hx^2(x^2+2xh+h^2)}=\dfrac{c(-2x-h)}{x^2(x^2+2xh+h^2)}=\\\\\\=\dfrac{-2xc-ch}{x^4+2x^3h+h^2x^2}$

Tomamos limite y tenemos que la derivada es

$\lim_{h\to 0}\dfrac{-2xc-ch}{x^4+2x^3h+h^2x^2}=\dfrac{-2xc}{x^4}=\dfrac{-2c}{x^3}$