## anonymous one year ago Use the following three steps to find and simplify the difference quotient of the function f(x)=\frac{1}{x-6}.

1. anonymous

f(x) = 1/x-6

2. anonymous

I believe I do the correct work but for the last two steps it says I am wrong.

3. anonymous

@ganeshie8

4. anonymous

@Hero

5. Jhannybean

The difference quotient: $$\dfrac{f(x+h)-f(x)}{h}$$ So let's begin by finding $$f(x+h)$$ first. Wherever you see an x, replace it with $$x+h$$

6. anonymous

1/x-6 = 1/(x+h)-6

7. anonymous

then i'm suppose to subtract it by f(x)

8. Jhannybean

Alright, now we plug it in to the formula. And you are already given $$f(x)$$.

9. anonymous

1/(x+h)-6 - 1/x = -h-6/x(x+h)-6

10. Jhannybean

$\frac{f(x+h)-f(x)}{h} = \frac{\dfrac{1}{x+h-6}-\dfrac{1}{x-6}}{h}$

11. Jhannybean

When I find a common denominator, I multiply both the denominators of the fraction together.

12. Jhannybean

$=\frac{\dfrac{x-6 -(x+h-6)}{(x-6)(x+h-6)}}{h}$

13. Jhannybean

Now let's simplify both the numerator and denominator.

14. anonymous

isn't the top suppose to be -h

15. anonymous

-h/(x+h-6)(x-6)

16. anonymous

then -h/(x+h-6)(x-6)/h

17. Jhannybean

Yep, you're on track. That's what I got too. $=\frac{\dfrac{x-6-x-h+6}{(x-6)(x+h-6)}}{h} = -\frac{\dfrac{h}{(x-6)(x+h-6)}}{h}$

18. Jhannybean

Now remember the rule of dividing fractions? $$\dfrac{\dfrac{a}{b}}{c} \iff \dfrac{a}{bc}$$

19. anonymous

its -h^2/(x+h-6)(x-6)

20. Jhannybean

We're going to apply this rule to our simplified function. We multiply $$(x-6)(x+h-6)$$ with $$h$$.

21. Jhannybean

Not exactly.

22. Jhannybean

$-\frac{\dfrac{h}{(x-6)(x+h-6)}}{h} = -\frac{h}{h(x-6)(x+h-6)}$Therefore both the h's cancel out.

23. Jhannybean

$-\frac{\cancel{h}}{\cancel{h}(x-6)(x+h-6)}$

24. anonymous

ahhh i understand

25. anonymous

it says the bottom is not correct

26. anonymous

nevermind

27. Jhannybean

when you think of a fraction in the form $$\dfrac{\dfrac{a}{b}}{c}$$ think of $$\dfrac{a}{b} \cdot \dfrac{1}{c} = \dfrac{a}{bc}$$

28. anonymous

i forgot the - lol

29. Jhannybean

Ahh, good good.