whats the derivative of 1/x?

Answer 1

I know what it is already, but we can prove what it is using the definition of a derivative.

f(x) = 1/x

f'(x) = lim [f(x + h) - f(x)]/h
h -> 0

f'(x) = lim [ 1/(x + h) - 1/x ]/h
h -> 0

Multiply top and bottom by x(x + h).

lim [ x - (x + h) ] / [hx(x + h)]
h -> 0

Simplify

lim [ -h / [hx(x + h)] ]
h -> 0

Cancel.

lim [ -1/[x(x + h)]
h -> 0

And now we can safely plug in h = 0.

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

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

Answer 2

1/x = x^(-1)
derivative is (-1) x^(-2) = -1/x^2

Answer 3

-1/x^2

Answer 4

1/x=x^-1
The derivative of x^n is nx^(n-1), so the derivative of 1/x=-1*x^(-2), or -1/x².