Question

find the derivative of g(x) = f(x)^3 by using the definition

Answer 1

\[\lim_{h\rightarrow0}\frac {f(x+h)-f(x)}{h} =\lim_{h\rightarrow0}\frac {(x+h)^3-x^3}{h} = \lim_{h\rightarrow0}\frac {(x^3+3hx^2+3h^2x+h^3)-x^3}{h} \]

You should be able to do the rest...

Answer 2

@whpalmer4, what you posted was the derivative of
\[f(x)=x^3,\]
whereas the question was finding the derivative of
\[g(x) = [f(x)]^3\]
using the limit definition.

Answer 3

According to the definition, the derivative of g(x) is
\[\lim_{h\to 0} \dfrac{[f(x+h)]^3-[f(x)]^3}{h},\]
which becomes much trickier to figure out.