Question

Find the derivative $x^3$

Answer 1

According to first principle:
$f^{\prime }\left(x\right)=\underset{h\to 0}{lim}\frac{f(x+h)-f\left(x\right)}{h}$
Here, $f\left(x\right)=x^3$
$(x^3{)}^{\prime }=\underset{h\to 0}{lim}\frac{(x+h{)}^3-x^3}{h}$
$=\underset{h\to 0}{lim}\frac{x^3+3x^{2}h+3x{h}^2+h^3-x^3}{h}$
$=\underset{h\to 0}{lim}\frac{3x^{2}h+3x{h}^2+h^3}{h}$
$=\underset{h\to 0}{lim}3x^2+3xh+h^2$
$=3x^2$