Question

Find the derivative of the following function from first principle.

x³ – 27

Answer 1

Let f(x) = x³ – 27. Accordingly, from the first principle,

`f'(x) = lim_(h->0)(f(x + h) - f(x))/h`

= `lim_(h->0)([(x + h)^3 - 27] - (x^3 - 27))/h`

= `lim_(h->0)(x^3 + h^3 + 3x^2h + 3xh^2 - x^3)/h`

= `lim_(h->0) (h^3 + 3x^2h + 3xh^2)/h`

= `lim_(h->0) (h^3 + 3x^2 + 3xh)`

= 0 + 3x² + 0

= 3x²