Question

Cuando se deriva f(x)= 2x^3 utilizando la expresion de limites resolviendo el producto notable y reduciendo terminos semejantes resulta la expresion:

a) lim h-->0 x^3+3xh-h^3-5
-------------------------
h

b)lim x-->0 6x^2h+6xh^2+2h^3
----------------------------
h

C) lim h-->0 2x^2h+2xh^2-2h^3
---------------------------
h

d) lim x-->0 2+6x^2+3xh-h^2
------------------------
h

Answer 1

SOLUCIÓN  

Hola‼

Recordemos la definición de derivada

                               $\boxed{\mathrm{\boldsymbol{f '(a) =\lim_{h \to 0} \frac{f(a+h)-f(a)}{h}}}}$

Entonces en el problema

                             $f '(x) =\lim_{h \to 0} \dfrac{f(x+h)-f(x)}{h}}\\\\\\f '(x) =\lim_{h \to 0} \dfrac{2(x+h)^{3}-2x^{3}}{h}}\\\\\\f '(x) =\lim_{h \to 0} \dfrac{2(x^{3} + 3x^{2}h+3xh^{2}+h^{3})-2x^{3}}{h}}\\\\\\f '(x) =\lim_{h \to 0} \dfrac{2x^{3} + 6x^{2}h+6xh^{2}+2h^{3}-2x^{3}}{h}}\\\\\\\boxed{\boldsymbol{\mathrm{f '(x) =\lim_{h \to 0} \dfrac{6x^{2}h+6xh^{2}+2h^{3}}{h}}}}}$