Calculas Question need assistance

Question

The point (1,2) is on the graph of:

x^2+3xy+y^4=x^2+22

Find the slope of the line tangent to the graph at that point:

dy/dx =

Answer 1

Let's use implicit differentiation to solve this. Differentiate each side of the equation with respect to x. Be sure to not forget product rule for the 3xy term.

d/dx (x² + 3xy + y⁴) = d/dx (x² + 22)

2x + 3y + 3x*dy/dx +4y³*dy/dx = 2x

Now, put all the dy/dx terms on one side of the equation

dy/dx(3x+4y³) = 3y

dy/dx = 3y/(3x+4y³)

Use the point (1,2) and plug it in to solve for dy/dx

dy/dx = 3*2/(3*1 + 4*2³) = 6/35