Differentiation of x^3: Find d/dx(x^3)

Answer

Using the definition of the derivative for $f(x)=x^3$,

$$f'(x)=\lim_{h\to0}\frac{(x+h)^3-x^3}{h}.$$

Expand the cube:

$$(x+h)^3=x^3+3x^2h+3xh^2+h^3.$$

Then

$$f'(x)=\lim_{h\to0}\frac{3x^2h+3xh^2+h^3}{h} =\lim_{h\to0}(3x^2+3xh+h^2).$$

Taking the limit gives

$$\boxed{\frac{d}{dx}(x^3)=3x^2}.$$

This also matches the power rule, $\frac{d}{dx}x^n=nx^{n-1}$.