Use implicit differentiation to find the slope of the tangent line to the curve (y/x+8y)=x^8 +9 at the point (1,10/-79).
I would rather use the product rule than the quotient rule, so lets multiply each term by x to get
y + 8xy = x^9 + 9x
dy/dx + 8x(dy/dx) + 8y = 9x^8 + 9
dy/dx(1+8x) = 9x^8 + 9 - 8y
dy/dx = (9x^8 + 9 - 8y)/(1+8x)
plug in your point and you are done.