Question

obtain the derivative of the function y=X2+1 by first principal (definition).​

Answer 1

By First Principal,

${f}^{ l} x \: = lim \: h→0 \: \frac{f(x + h) - f(x)}{h} \\ = lim \: h→0 \: \frac{ {((x + h)}^{2} + 1) - ( {x}^{2} + 1)}{h} \\ = lim \: h→0 \: \frac{( {x}^{2} + {h}^{2} + 2xh + 1 - {x}^{2} - 1) }{h} \\ = lim \: h→0 \: \frac{( {h}^{2} + 2xh) }{h} \\ = lim \: h→0 \: (h + 2x) \\ = (0 + 2x) = 2x$

Using formulas,

$\frac{dy}{dx} = \frac{d( {x}^{2} + 1) }{dx} = \frac{d( {x)}^{2} }{dx} + \frac{d(1)}{dx} \\ = 2 {x}^{2 - 1} + 0 = 2x$

Hope It Helps :)