Evaluate the difference quotient for the given function. Simplify your answer.
f(x)=1/x, (f(x)-f(a))/(x-a)
(f(x)-f(a))/(x-a)
= (1/x - 1/a)/(x-a)
= ( (a - x)/(ax) )/(x-a)
now (a-x)/(x-a) = -1
so ..
= -1/(ax)
To evaluate the difference quotient for the function f(x) = 1/x, we'll use the formula:
(f(x) - f(a))/(x - a)
Let's substitute the given function f(x) = 1/x into this formula:
(f(x) - f(a))/(x - a) = (1/x - f(a))/(x - a)
To simplify this fraction, we need to find a common denominator. The common denominator in this case is x(x - a).
(f(x) - f(a))/(x - a) = (1/x - f(a))/(x - a) * (x/x) = (1 - x*f(a))/(x(x - a))
The simplified difference quotient for the function f(x) = 1/x is:
(1 - x*f(a))/(x(x - a))