Question

Find an equation of the line tangent to the curve √xy + 4y = 39 at the point (1,9)?

Answer 1

Find the derivative of the curve to find the line slope at the given point

sqrt(xy) + 4 dy/dx = 39

0.5(xy)^(-0.5) * (y + x dy/dx) + 4dy/dx=  0

No need to amplify simply plug in the given point ( 1,9)

0.5 ( 1 * 9)^(-0.5)* ( 9 + dy/dx) + 4dy/dx  = 0 

Solve for dy/dx

1/6( 9  + dy/dx)  = -4dy/dx

9 + dy/dx = -24 dy/dx

-25dy/dx = 9

dy/dx = -9/25

as your question implies , you know the rest of the steps.