Answer:
Option b. 12
Step-by-step explanation:
This exercise asks us to find the derivative of a function using the definition of a derivative.
Our function is [tex]f(x) = x^{3}[/tex]. Therefore:
[tex]f(2+h) = (2+h)^{3}[/tex]
[tex]f(2) = (2)^{3} = 8[/tex]
Then:
[tex]\lim_{h \to \0} \frac{f(2+h)-f(2)}{h}=\lim_{h \to \0} \frac{(2+h)^{3}-8}{h}[/tex]
Expanding:
[tex]\lim_{h \to \0} \frac{(2+h)^{3}-8}{h} =\lim_{h \to \0} \frac{8+ h^{3} +6h(2+h) -8}{h} =\lim_{h \to \0} \frac{h^{3} +6h(2+h)}{h}[/tex]
[tex]\lim_{h \to \0} \frac{h^{3}+ 6h(2+h)}{h} =\lim_{h \to \0} h^{2} + 6(2+h) [/tex]
Now, if x=0:
[tex]\lim_{h \to \0} \frac{f(2+h)-f(2)}{h} = (0)^{2} +6(2+0) = 12[/tex]
