Final answer:
To find the equation of the lines tangent and normal to the function x^2y + yx = 2x using implicit differentiation, we need to find the first derivative of the equation and substitute a given point to get the equations of the tangent and normal lines.
Step-by-step explanation:
To find the equation of the lines tangent and normal to the function x^2y + yx = 2x using implicit differentiation, we need to find the first derivative of the equation. Differentiating both sides with respect to x, we get 2xy + x^2(dy/dx) + y = 2. Solving for dy/dx, we get dy/dx = (2 - 2xy)/(x^2 + 1).
To find the equation of the tangent line, we substitute the given point (x₀, y₀) into the equation dy/dx. Then, we use the point-slope form y - y₀ = m(x - x₀), where m is the slope dy/dx. This gives us the equation of the tangent line. To find the equation of the normal line, we use the fact that the slope of a normal line is the negative reciprocal of the slope of the tangent line. So, the equation of the normal line can be obtained using the same point-slope form but with the negative reciprocal slope.