Find the difference quotient f(x+h)-f(x)/h, where h cannot be equal to 0 , for the function below.
f(x)=1/x+9
Simplify your answer as much as possible.
If that is 1/(x+9), then just plug and chug:
f(x+h)-f(x) = 1/(x+h+9) - 1/(x+9)
= [(x+9)-(x+h+9)]/[(x+h+9)(x+9)]
= (-h)/[(x+h+9)(x+9)]
That is the difference. The difference quitient is just that divided by h, so it is
-1/[(x+h+9)(x+9)]
To find the difference quotient for the given function f(x) = 1/x + 9, we need to substitute f(x + h) and f(x) into the formula (f(x + h) - f(x))/h, where h is not equal to 0.
Step 1: Find f(x + h)
Substitute (x + h) into the function:
f(x + h) = 1/(x + h) + 9
Step 2: Find f(x)
Substitute x into the function:
f(x) = 1/x + 9
Step 3: Calculate f(x + h) - f(x)
Substitute the values obtained in steps 1 and 2 into the formula:
f(x + h) - f(x) = (1/(x + h) + 9) - (1/x + 9)
Step 4: Simplify the expression
To combine the fractions, we need a common denominator. The common denominator is (x(x + h)):
f(x + h) - f(x) = (1/(x + h) * x/x) - (1/x * (x + h)/(x + h)) + 0
Now simplify the expression:
f(x + h) - f(x) = (x - (x + h))/(x(x + h)) + 0
Use the distributive property to simplify further:
f(x + h) - f(x) = (x - x - h)/(x(x + h))
Finally, cancel out the like terms:
f(x + h) - f(x) = -h/(x(x + h))
Step 5: Divide by h
To get the difference quotient, divide the expression obtained in step 4 by h:
(f(x + h) - f(x))/h = (-h/(x(x + h)))/h
Step 6: Simplify
Dividing by h on the numerator and denominator, we get:
(f(x + h) - f(x))/h = -1/(x(x + h))
Therefore, the difference quotient for the function f(x) = 1/x + 9 is -1/(x(x + h)).