Answer
The behaviors of the curves in the plotted graph show that the derivative of $f(x)=x^3$ is $3x^2$.
Work Step by Step
The graphs of $y=3x^2$ and $y=\frac{(x+h)^3-x^3}{h}$ are plotted and denoted below.
We see that as $h$ gets smaller and smaller and approaches $0$, the curve of $y=\frac{(x+h)^3-x^3}{h}$ gets closer and closer from both the left and right and becomes the same as the curve of $y=3x^2$ in the center..
We can thus deduce that: $$\lim_{h\to0}\frac{(x+h)^3-x^3}{h}=3x^2$$
According to the definition of derivative, $\lim_{h\to0}\frac{\sqrt{x+h}-\sqrt x}{h}=f'(x)$ for $f(x)=x^3$.
Therefore, the behaviors of the curves in the plotted graph show that the derivative of $f(x)=x^3$ is $3x^2$.
