Question

derivative of x+1/x by first principle​

Answer 1

Answer:

$\frac{df}{dx} = lim_{h \to0} \frac{f(x + h) - f(x)}{h} \\ \\ = lim_{h \to0} \frac{x + h + \frac{1}{x + h} - x - \frac{1}{x} }{h} \\ \\ = lim_{h \to0} \frac{h(x + h)x -hx - h(x + h) }{hx(x + h)} \\ \\ = lim_{h \to0} \frac{h {x}^{2} + {h}^{2}x - hx - hx - {h}^{2} }{ {h}^{2}x + h {x}^{2} } \\ \\ = lim_{h \to0} \frac{ {h}^{2} (x - 1) + ( {x}^{2} -2x )h }{x {h}^{2} + {x}^{2} h} \\ \\ = \frac{x - 1}{x}$