3x2y2 + 4y5 + 8x2y3 + xy = 5

Question

Find the derivative of each expression using implicit differentiation.

Answer 1

Never mind...Jon's got this one.

Answer 2

3x²y² + 4y⁵ + 8x²y³ + xy = 5

 

Take the derivative of both sides:

D_x 3x²y² + 4y⁵ + 8x²y³ + xy = D_x 5

 

Find the derivative of each term.  The product rule and the chain rule will have to be used.  Use y' to represent dy/dx, so it's easier to write out:

6x y² + 3x² (2y) y' + 20y⁴ y' + 16x y³ + 8x² (3y²) y' + y + xy' = 0

 

Gather terms with y' and factor out y':

y' (6x²y + 20y⁴ + 24x²y² + x) + (6xy² + 16xy³ + y) = 0

y' (6x²y + 20y⁴ + 24x²y² + x) = -(6xy² + 16xy³ + y)

y' = -(6xy² + 16xy³ + y) / (6x²y + 20y⁴ + 24x²y² + x)

y' = -y (6xy+ 16xy² + 1) / (6x²y + 20y⁴ + 24x²y² + x)

 

 

That looks to be about as far as you can get.