Determine the derivative

Question

What is the derivative of this? How do I find it?

Answer 1

Since you have x and y mixed together, you should use implicit differentiation. The first term is the product rule.

x(1/y)y'+ln y+3y²y' = -10x^-3 You have to get all the y' terms on one side, so subtract ln y from both sides

x(1/y)y'+3y²y' = -10x^-3 - ln y Now factor out y' from each term

y' [(x/y)+3y² ] = -10x^-3 - ln y Now divide both sides by everything in the left in the brackets

y' = [-10x^-3 - ln y] / [(x/y)+3y² ]

They may want the final answer in a more simplified form. Depends on your instructor I think!

Answer 2

Use implicit differentiation

xlny + y³ = 5x^-2

lny + x/y(dy/dx) + 3y²(dy/dx) = -10/x³

dy/dx(x/y + 3y²) = -lny - 10/x³

dy/dx = [-lny - 10/x³] / (x/y + 3y²)