Use implicit differentiation to find the slope of the tangent line to the curve (y/x+6)=x^9-7) pt. (1,-6/37)

I've tried to solve for the derivative of (y/x+6y) using quotient and chain rule but cant seem to get past that step, and when I try to find M (slope of tangent line) at the point (1,-6/37) I cannot find the right answer...help please!

In the heading you have (y/x+6), but further down you have (y/x+6y)
I'll assume the latter, since this is the only case that gives us (1, -6/37)
Also, that point is on the curve only when all of x+6y is in denominator.
So write equation as:

y/(x+6y) = x⁹ - 7

Differentiate implicitly:
(y' (x+6y) - y (1 + 6y')) / (x+6y)² = 9x⁸
xy' + 6yy' - y - 6yy' = 9x⁸(x+6y)²
xy' = 9x⁸(x+6y)² + y
y' = 9x⁷(x+6y)² + (y/x)

At point (1, -6/37), slope is
y' = 9 (1 + 6(-6/37))² + -6/37
y' = 9 (1 - 36/37) - 6/37
y' = 9 (1/37)² - 6/37
y' = 9/37² - (6*37)/37²
y' = -213/1369

Equation of tangent line:
y + 6/37 = -213/1369 (x - 1)
y = -213/1369 x + 213/1369 - 6/37
y = -213/1369 x - 9/1369

Check:
http://www.wolframalpha.com/input/?i=y%2…

Hey it's Ali Baba haha

I believe I can help :P

so I used the Quotient Rule and got dy/dx(x+6y)-y(1+6 dy/dx)/(x+6y)^2=9x^8.

After that I multiplied the denominator (x+6y)^2 to the other side to get rid of it then got dy/dx(x+6y)-y(1+6dy/dx)=9x^8(x+6y)^2. Then you distribute "y(1+6dy/dx)" and add "y" to the other side because you want to have all y primes on one side and the rest on the other side so once you add "y" and simplify the rest you get dy/dx(x+6y)=9x^8(x+6)^2+y.

Finally you get dy/dx= [9x^8(x+6y)^2+y]/x

after doing implicit differentiation you plug in your points (1, -6/37) to your final answer and bam! lol there's your slope aka M :D

ps. if you're plugging this in to webworks try to not put it in the calculator to get a numerical answer instead plug in your final equation followed by your points and webworks automatically solves it for you when you submit your answer. :)