SOLUTION: Suppose f(x)=1/x. Write an expression in terms of x and h that represents the average rate of change of f over any interval of length h. [That is, over any interval (x,x+h).] Simpl

Question 1149041: Suppose f(x)=1/x. Write an expression in terms of x and h that represents the average rate of change of f over any interval of length h. [That is, over any interval (x,x+h).] Simplify your answer as much as possible.
Answer by rothauserc(4718) (Show Source): You can put this solution on YOUR website!
The problem is asking the first derivative of f(x) using the limit definition of the first derivation, that is,
:
f'(x) = limit as h approaches 0 of (f(x+h) -f(x))/h
:
We are given f(x) = 1/x
:
f(x+h) = 1/(x+h)
:
f'(x) = limit as h approaches 0 of ( 1/(x+h) - 1/x) )/h =
:
(x - (x+h) ) / ( h * x * (x+h)) =
:
-h/(h * x * (x+h)) =
:
-1/(x^2 +xh)
:
h approaches 0, so we are left with
:
f'(x) = -1/x^2
:

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