Question

Find the derivative of the following functions from first principle
$x^3-27$

Answer 1

Let $f\left(x\right)=x^3-27$
According to the first principle,
$f^{{}^{\prime }}\left(x\right)=\underset{h\to 0}{lim}\frac{f(x+h)-f\left(x\right)}{h}$
= $\underset{h\to 0}{lim}\frac{[{(x+h)}^3-27]-(x^3-27)}{h}$
= $\underset{h\to 0}{lim}\frac{x^3+h^3+3x^{2}h+3x{h}^2-x^3}{h}$
= $\underset{h\to 0}{lim}\frac{h^3+3x^{2}h+3x{h}^2}{h}$
= $\underset{h\to 0}{lim}(h^2+3x^2+3xh)$
= $0+3x^2+0=3x^2$