How do you differentiate #y=(6+7x)/(14x+9)#?

This is just a basic quotient rule problem. The general formula for quotient rule is as follows:

For #f(x)/g(x)#:

#d/dx(f(x)/g(x)) = (d/dx[f(x)]*g(x) - d/dx[g(x)]*f(x))/(g(x))^2#

Where #f(x)# is the function in your numerator, and #g(x)# is the function in your denominator.

Now, all we have to do is plug in everything.

#=> (d/dx(6+7x)(14x+9) - d/dx(14x+9)(6+7x))/(14x+9)^2#

The derivatives you need to take here are quite simple, as both of your functions are linear. In any case, let's take them up:

#d/dx(6+7x) = 7#

#d/dx(14x+9) = 14#

Now we just plug this back:

#=> (7(14x+9) - 14(6+7x))/(14x+9)^2#

Usually, it's best to leave this answer as is, as expanding doesn't get you anywhere. In this case, however, let's expand the numerator:

#=> (98x+63-84-98x)/(14x-9)^2#

As you can see, the #98x#'s cancel out. So, we're left with:

#=>-21/((14x-9)^2#

And that is your final answer!

If you're struggling with the Product and Quotient rule, I'd recommend watching this video by Professor Leonard. It's a long video, but he really talks about it thoroughly, and it will help you understand the concepts better.

Hope that helped :)