How do you find the formula for the derivative of $1/x$ ?

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Answer 1 · 2015-04-14T18:07:15

$(d x^a)/(dx) = a*x^(a-1)$

$1/x = x^(-1)$

Therefore
$(d 1/x)/(dx) = -x^(-2)$ or $- 1/(x^2)$

Answer 2 · 2015-04-14T18:14:14

I will assume that you are working from first principles, that is the definition.

There are two choices for how to "officially" define the derivative, every textbook author and teacher makes a decision which one to make official.

I'll use: $f'(x)=lim_(hrarr0)(f(x+h)-f(x))/h$

For $f(x) = 1/x$ , we get:

$f'(x) = lim_(h rarr0) (f(x+h)-f(x))/h$

$color(white)"ssssss"$ $= lim_(h rarr0) (1/(x+h)-1/x)/h$

$color(white)"ssssss"$ $= lim_(h rarr0) ((x-(x+h))/(x(x+h)))/(h/1)$

$color(white)"ssssss"$ $= lim_(h rarr0) ((x-x-h))/(x(x+h))*(1/h)$

$color(white)"ssssss"$ $= lim_(h rarr0) ((-1))/(x(x+h))$

$color(white)"ssssss"$ $= -1/x^2$

That is: $f'(x) = -1/x^2$

How do you find the formula for the derivative of 1/x1/x?