Find the slope of the tangent to the curve y^2=x^3/(xy+6) at (6,3)

Answer 1

Take the implicit derivative, solve for the derivative term y', and plug in x and y to find the slope at the point.

Answer 2

First, take the derivative of the function (this will require implicit differentiation, quotient rule, and product rule):

2y(dy/dx)= [ (xy+6)(3x^2) - (x^3)(x(dy/dx)+y) ] / [ (xy+6)^2 ]

Now plug in the values you've been given for x and y (x=6, y=3) and simplify to isolate dy/dx.

I got dy/dx=1944/4752 which simplifies to 9/22. So the slope of the tangent line is m=9/22

Now to write the final equation use point-slope form- y-y1=m(x-x1), plug in m, y1, and x1 and simplify to get the final equation for the tangent line.