Question

Find the derivative of the following function from first principles.
y=x^n​

Answer 1

Answer

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,

$\sf f(x) = {x}^{n}$

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

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

$\displaystyle \sf f (x) = \lim_{h\to 0} \frac{{x}^{n} + n. {x}^{n - 1} + ... + {h}^{n} - {x}^{n} }{h}$

$\displaystyle \sf f (x) = \lim_{h\to 0} \frac{h( n. {x}^{n - 1} + ... + {h}^{n - 1} )}{h}$

$\displaystyle \sf f (x) = \lim_{h\to 0} ( n. {x}^{n - 1} + ... + {h}^{n - 1} )$

$\displaystyle \sf f (x) = n. {x}^{n - 1}$

thus,

$\sf \dfrac{d}{dx} {x}^{n} = n. {x}^{n - 1}$