Question

find the derivate of x^3 - 27 using the first principle​

Answer 1

The first principle of derivative :

Suppose f is a real value function defined by $\displaystyle \sf {f}^{1} (x) = \lim_{h\to 0} \frac{f(x + h) - f(x)}{h}$ where ever the limit exists is called derivative of f at x

According to the question,

$\sf f(x) = {x}^{3} - 27$

$\displaystyle \sf f (x) = \lim_{h\to 0} \dfrac{f(x + h) - f(x)}{h}$

$\displaystyle \sf f (x) = \lim_{h\to 0} \dfrac{(x + h)^{3} -27 - ( {x}^{3} - 27)}{h}$

$\displaystyle \sf f (x) = \lim_{h\to 0} \dfrac{ {x}^{3} + {3hx}^{2} + {3h}^{2}x + {h}^{3} - 27 - {x}^{3} + 27 }{h}$

$\displaystyle \sf f (x) = \lim_{h\to 0} \dfrac{3h {x}^{2} + {3h}^{2} x + {h}^{3} }{h}$

$\displaystyle \sf f (x) = \lim_{h\to 0} \dfrac{h(3{x}^{2} + {3}hx + {h}^{2}) }{h}$

$\displaystyle \sf f (x) = \lim_{h\to 0} (3{x}^{2} + {3}hx + {h}^{2})$

$\displaystyle \sf f (x) = 3{x}^{2}$