Question

Find the derivatives of $f\left(x\right)=x^3.$

Short answer

The derivative of $f\left(x\right)=x^3$is$3x^2.$

Step-by-step solution

Given Information

The given expression is$f\left(x\right)=x^3.$

Simplification 

$$\begin{gathered} f\text{'}\left(x\right)=\underset{h\to 0}{\lim }\frac{f(x+h)-f\left(x\right)}{h} \\ =\underset{h\to 0}{\lim }\frac{{(x+h)}^3-x^3}{h} \\ =\underset{h\to 0}{\lim }\frac{x^3+3x^{2}h+3x{h}^2+h^3-x^3}{h} \\ =\underset{h\to 0}{\lim }\frac{(3x^2+3xh).h}{h} \\ =\underset{h\to 0}{\lim }(3x^2+3xh) \\ =3x^2+3x.0 \\ =3x^2 \end{gathered}$$

Therefore, the derivate of$f\left(x\right)=x^3$is$3x^2.$