Difference Quotient of 2x/x-1
2(x+h)/(x+h)-1 - 2x/x-1
all divided by h
2x+2h/x+h-1 - 2x/x-1
all divided by h
2x cancels out
x cancels out
1 cancels out
(2h/h)/h
2h cancels with h
result: 2/h
basically you do the same as the previous question you had.
[f(x+h)-f(x)]/h
f(x+h)= [2(x+h)]/(x+h-1)
f(x)= 2x/(x-1)
[[2(x+h)]/(x+h-1)]-[2x/(x-1)]/h
then simplify! (hint: find a common denominator in order to combine to one whole fraction)
Make sure you put parenthesis or brackets to distinguish your numerator from your denominator. You can only cancel the numerator and denominator if the equation is one fraction under common denominator.
Let me try one more time
2x+2h/x+h-1 - 2x/x-1
---------------
h
2h and h cancel
2x+2/x-1 - 2x/x-1
---------------
h
2x+2-2x/x-1
---------------
h
2x cancel out
2/x-1
---------------
h
result: 2/xh-h
You have to remember you have to divide the "entire" numerator. You cannot just divide an h to one part of numerator to cancel. You can only cancel when the factors in the numerator are multiplying. For example, [(2x)(h)]/h. Your x+h-1 and x-1 belongs in the denominator!
It should look like this:
[(2x+2h)/[(x+h-1)(h)]] - [(2x)/[(x-1)(h)]]
Now find a common denominator:
This is my numerators:
______(h)(x+h-1) =(x-1)(h)______
How do I make it equal? What would I multiply on each side to make it equal?
To do this you
Want:
(x+h-1)(h)(x-1)
Relate:
[(2x+2h)/[(x+h-1)(h)]]
To get my want in the denominator, I need (x-1). Therefore, I multiply both the numerator and denominator by (x-1).
[(2x)/[(x-1)(h)]]
To get my want in the denominator, I need (x+h-1). Therefore, I multiply both the numerator and denominator by (x+h-1).
You should end up with:
[[(2x+2h)(x-1)]/[(x+h-1)(h)(x-1)]] - [[(2x(x+h-1)])/[(x+h-1)(h)(x-1)]]
Then combine fractions, since both share a common denominator:
[[(2x+2h)(x-1)-(2x(x+h-1)]]/[(x+h-1)(h)(x-1)]]
Then simplify!
Typo:
This is my "denominators*":
______(h)(x+h-1) =(x-1)(h)______
To find the difference quotient of 2x/(x-1), we first need to understand what the difference quotient represents. The difference quotient is the rate of change of a function as the input (in this case, x) changes by a small amount (in this case, h). It measures how the function's output changes with respect to the change in its input.
To compute the difference quotient, we start by plugging in the value of x+h into the function and subtracting the value of the function at x, then divide the whole expression by h.
So, we have:
2(x+h)/(x+h) - 1 - 2x/(x-1) divided by h
To simplify this expression, we will first simplify the two fractions separately.
1. Simplifying the first term: 2(x+h)/(x+h) - 1
Since (x+h)/(x+h) equals 1 for any non-zero value of h, we can simplify it to just 2 - 1.
2. Simplifying the second term: 2x/(x-1)
We can't directly simplify this fraction because the denominators are different.
Next, we subtract the second term from the first term:
(2 - 1) - 2x/(x-1)
To combine the two terms, we need a common denominator. In this case, the common denominator is (x-1). So, we multiply the first term by (x-1)/(x-1):
((2(x-1) - 1(x-1))/(x-1)) - 2x/(x-1)
Now, we can simplify the numerator:
(2x - 2 - x + 1 - 2x)/(x-1) - 2x/(x-1)
Simplifying further:
(-x - 1)/(x-1) - 2x/(x-1)
To add or subtract fractions, we need a common denominator. In this case, the common denominator is (x-1). So, we multiply the first term by (x-1)/(x-1):
((-x - 1)(x-1) - 2x)/(x-1)
Expanding the numerator:
((-x^2 + x - x + 1) - 2x)/(x-1)
Combining like terms:
(-x^2 - 2x + 1 - 2x)/(x-1)
Simplifying further:
(-x^2 - 4x + 1)/(x-1)
Finally, we divide the whole expression by h:
(-x^2 - 4x + 1)/(x-1) divided by h
To compute the limit as h approaches 0 (which is necessary to find the difference quotient), we simplify the expression without the h term:
(-x^2 - 4x + 1)/(x-1)
This expression represents the difference quotient of 2x/(x-1).