Find the derivative of (x^3 - 27) from first principle. - Vidyadip

Question

Find the derivative of $(x^3 - 27)$ from first principle.

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Answer

We have, $f(x) = x^3 - 27$
By using first principle of derivative,
${f^\prime }(x) = \mathop {\lim }\limits_{h \to 0} \frac{{f(x + h) - f(x)}}{h}$
$\therefore \quad {f^\prime }(x) = \mathop {\lim }\limits_{h \to 0} \frac{{\left[ {{{(x + h)}^3} - 27} \right] - \left[ {{x^3} - 27} \right]}}{h}$
$ = \mathop {\lim }\limits_{h \to 0} \frac{{{x^3} + 3{x^2}h + 3{h^2}x + {h^3} - 27 - {x^3} + 27}}{h}$
$ = \mathop {\lim }\limits_{h \to 0} \frac{{3{x^2}h + 3x{h^2} + {h^3}}}{h}$
$ = \mathop {\lim }\limits_{h \to 0} (3x^2 + 3xh + h^2) = 3x^2$

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