Question

[50б] найти производную функции
f(x)= x^(1/2)
! (по формуле через предел)!

на фото формула, по которой нужно решить

Answer 1

$\displaystyle \lim_{h \to0}\dfrac{f(x+h)-f(x)}{h}=\lim_{h \to0}\dfrac{(x+h)^{1/2}-x^{1/2}}{h}=\\ \\ \\ =\lim_{h \to0}\dfrac{((x+h)^{1/2}-x^{1/2})((x+h)^{1/2}+x^{1/2})}{h((x+h)^{1/2}+x^{1/2})}=\lim_{h \to0}\dfrac{x+h-x}{h(x^{1/2}+x^{1/2})}=\\ \\ \\ =\lim_{h \to0}\dfrac{h}{h\cdot 2x^{1/2}}=\frac{1}{2x^{1/2}}$