Find dy/dx for the relation 2xy^2+3*ln(y)=x^2-3y^3 at the point (3, 1).

I got 2/21 as the answer but the book says 1/6. Am I right? Can you show complete work so I see the procedure?

Tutors, sign in to answer this question.

2xy^2+3*ln(y)=x^2-3y^3

Differentiate both sdies with respect to x,

2(y^2 + 2xyy') + 3y'/y = 2x-9y^2y'

Plug in (3, 1),

2(1+6y')+3y' = 6-9y'

Solve for y',

24y' = 4

y' = 1/6 <==Answer

------

Attn: To get it done simpler, plug in (3,1) before you solve y' as a function of x.

Eric O. | Algebra 1, 2, Geometry, Pre-Calculus, CalculusAlgebra 1, 2, Geometry, Pre-Calculus, Ca...

This is an implicit differentiation problem.

2*x*y^2 + 3*LN(y) = x^2 - 3*y^3

differentiating using the product rule and chain rule for the first half of the left side, and the chain rule for the second half of the left side yields

4xyy' + 2y^2 + (3/y)y'

for the first half of the right side we differentiate as usual and for the last half we use the chain rule, which yields

2x - (9y^2)y'

the whole thing being

4xyy' + 2y^2 + (3/y)y' = 2x - (9y^2)y'

grouping like terms and solving for y' (dy/dx) yields

dy/dx = (2x -2y^2) / (4xy + (3/y) + 9y^2)

Plugging in or point at (3,1) yields

dy/dx = (6 - 2) / (12 + 3 + 9) = 4/24 = 1/6

Already have an account? Log in

By signing up, I agree to Wyzant’s terms of use and privacy policy.

Or

To present the tutors that are the best fit for you, we’ll need your ZIP code.

Your Facebook email address is associated with a Wyzant tutor account. Please use a different email address to create a new student account.

Good news! It looks like you already have an account registered with the email address **you provided**.

It looks like this is your first time here. Welcome!

To present the tutors that are the best fit for you, we’ll need your ZIP code.

Please try again, our system had a problem processing your request.

Nicole H.

Applied Math Master's Student with 3 Years of Teaching Experience

$10 per 15 min

View Profile >

Corey G.

I make math easy to understand

$11.25 per 15 min

View Profile >

Sotirios K.

Physics and Math tutor to help you reach your educational goals

$11.25 per 15 min

View Profile >