Differential calculus problem using limits

[Guest]

1.a) Use the formal definition of the derivative to compute f'(2) when
[tex]f(x)=(1-x)/2x[/tex]

1.b) Do the same to find f'(x)

[nathi123]

a) [tex]f'(2) = \lim_{x \to 2}\frac{f(x) - f(2)}{x-2}=\lim_{x \to 2}\frac{\frac{1-x}{2x}-\frac{-1}{4}}{x-2}=\lim_{x \to 2}\frac{2-x}{4x(x-2)}=\lim_{x \to 2}(-\frac{1}{4x})=-\frac{1}{8}[/tex]
b) f'(x) = [tex]\lim_{x \to x_{0 }}\frac{f(x) - f(x_{0 }}{x-x_{0 }}=\lim_{x \to x_{0 }}\frac{\frac{1-x}{2x}-\frac{1-x_{0 }}{2x_{0 }}}{x-x_{0 }}=\lim_{x \to x_{0 }}\frac{x_{0 }-x}{2xx_{0 }(x-x_{0 })}=\lim_{x \to x_{0 }}\frac{-1}{2xx_{0}}=-\frac{1}{2x_{0 }^{2}}[/tex].

[Guest]

Could you explain the formulas used and why?

[nathi123]

I use the difinition of the derivative in point ,if y = f(x) [tex]\Rightarrow y'(x_{0}) = \lim_{x \to x_{0 }}\frac{f(x)-f(x_{0})}{x-x_{0 }}[/tex]