Question

Using definition of derivative prove that (1/x)'=-1/x^2

Answer 1

Example 2: (d/dx)(1/x) = -1/(x2)

f(x) = 1/x

f'(x) = (d/dx)(1/x) = (d/dx)(x-1) = (-1)x-2 = -1/(x2)

Answer 2

(1/x)' = -1/x^2

D(1/x) = -1/x^2

x(0) - 1(1)
--------- = -1/x^2
x^2

-1/x^2 = -1/x^2

Answer 3

\[f'(x)=\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}
=\lim_{h\rightarrow 0}\frac{\frac{1}{x+h}-\frac{1}{x}}{h}=
\lim_{h\rightarrow 0}\frac{-1}{x(x+h)}=-\frac{1}{x^2}\]

Answer 4

hmmm.... i spose that would be more accutate depending on your definition of a derivative :)