How do you find the derivative of 1/x using limit definitions?

The answer is as follows:

f(x) = 1/x
f(x + h) = 1/(x + h)

(f(x + h) - f(x)) / (x + h - x) =>
(1/(x + h) - 1/x) / h =>
((x - (x + h)) /(x * (x + h))) / h =>
(x - x - h) / (h * x * (x + h)) =>
-h / (h * x * (x + h)) =>
-1 / (x * (x + h))

h goes to 0.

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

The same way you find every derivative of a function.

Definition: f’(x) = lim h → 0 { [ f(x+h) - f(x) ] / h }

f’(x) = lim h→0 { [ f(x+h) - f(x) ] / h }

f( x ) = 1/x
f( x+h ) = 1/( x+h ) 

f’(x) = lim h→0 { [ 1/(x+h) - ( 1/x ) ] / h }
f’(x) = lim h→0 { [ ( x - x - h ) / ( x² + xh )  ] / h }
f’(x) = lim h→0 { [ ( -h ) / ( x² + xh ) ] / h }
f’(x) = lim h→0 { -1 / ( x² + xh )  }
f’(x) = -1 / ( x² + x * 0 ) 
f’(x) = -1 / x²
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