Expert: Sherry Wallin - 12/17/2010

Question
please help me solve, step by step, f(x)= 20/x+15 in the form f(x+h)-f(x)/ h. thank you

Answer
lexisha~

You need to write what you want properly. Do you mean (20/x) + 15? I doubt it, but yet, what you wrote is equivalent to that. I think what you meant is 20/(x+15) and thus you must use the parenthesis appropriately. I do know you want [f(x+h)-f(x)]/h but what you have written is equivalent to f(x+h) -[f(x)/h]. So please be careful and communicate carefully :)

Break up what you want into smaller pieces:
f(x+h) = 20/[x+h+15]  
and f(x) = 20/(x+15)
and you will be multiplying the whole expression by (1/h)

thus [f(x+h)-f(x)]/h = [20/[x+h+15] - 20/(x+15)]/h Notice that dividing by h is like multiplying by 1/h thus you really have (1/h)[20/[x+h+15] - 20/(x+15)] and hold (1/h) off to the side and then

Now here you will need to find a common denominator and I have a few tricks up my sleeve, it is called whip, whip, whap and it is a way to add fractions with different denominators without finding that common denominator and it goes like this a/b + c/d = (ad + bc)/bd, in words take the product of the first numerator and the second denominator and add it to the product of the first denominator and the second numerator, the sum of this product divided by the product of the denominators. Example: 2/3 + 1/5 = [2(5)+ 3(1)]/(3(5)) = 13/15

Now back to our problem:
[20(x+15) - 20(x+h+15)]/[(x+h+15)(x+15)]
= [20x + 300 -20x -20h - 300]/[(x+h+15)(x+15)]  
= -20h/[(x+h+15)(x+15)]  now don't forget you still have a factor of 1/h:
= (1/h)(-20h/[(x+h+15)(x+15)])
= -20/[(x+h+15)(x+15)]  you can expand the demoninator but it is not necessary    

Math Prof