Find the difference quotient for the following Rational Function f(x)=5/x using the four steps outlined below.
a. find f(x+h)
b. find f(x)
c. substitute f(x+h) and f(x) into the difference quotient formula.
d. simplify
f(x+h) = 5/(x+h)
f(x) = 5/x
[f(x+h) - f(x)]/h = [5x -5(x+h)]/[hx(x+h)]
= -5 /[x(x+h) }
= -5/(x^2+hx)
By the way,
if you take the limit of that as h--->0
then you are a calculus student and you just found the derivative
d/dx (5/x) = -5/x^2
To find the difference quotient for the rational function f(x) = 5/x using the four steps outlined, follow these instructions:
a. Find f(x + h)
1. Replace x with (x + h) in the function f(x): f(x + h) = 5/(x + h)
b. Find f(x)
2. Use the original function f(x): f(x) = 5/x
c. Substitute f(x + h) and f(x) into the difference quotient formula
3. The difference quotient formula is given by:
(f(x + h) - f(x))/h
So, substitute the values of f(x + h) and f(x) into the formula:
((5/(x + h)) - (5/x))/h
d. Simplify
4. To simplify the expression, we need to combine the terms:
((5x - 5(x + h))/(x(x + h)))/h
Simplify further:
(5x - 5x - 5h)/(x(x + h))/h
-5h/(x(x + h))/h
Finally, cancel out the h terms:
-5/(x(x + h))
Therefore, the difference quotient for the rational function f(x) = 5/x is -5/(x(x + h)).