Question

How do you use implicit differentiation to find `(dy)/(dx)` given `5x^3+xy^2=5x^3y^3`?

Answer 1

1. Differentiate each term
   `d(5x^3)+d(xy^2)=d(5x^3y^3)`

2. Pull constants out of each differential
   `5d(x^3)+d(xy^2)=5d(x^3y^3)`

3. Differentiate
   `5(color(red)(3x^2dx))+[color(blue)(x(2y)dy+y^2dx)]=5[color(purple)(x^3(3y^2)dy+y^3(3x^2)dx)]`

4. Simplify
   `(15x^2)dx+(2xy)dy+(y^2)dx=(15x^3y^2)dy+(15x^2y^3)dx`

5. Separate `dy` and `dx` terms
   `(2xy)dy-(15x^3y^2)dy=(15x^2y^3)dx-(15x^2)dx-(y^2)dx`

6. Simplify terms
   `(-15x^3y^2+2xy)dy=(15x^2y^3-15x^2-y^2)dx`

7. Solve for `dy/dx`
   `dy/dx=(15x^2y^3-15x^2-y^2)/(-15x^3y^2+2xy)`