How Is Implicit Differentiation Applied to Solve 5y^2 = 4x - 3/4x + 3?

[Jan Hill]

Homework Statement

Given 5y^2 = 4x - 3/4x + 3

Homework Equations

is it permissable to say this is equal to y^2 = 4x -3 /5(4x + 3) and then 2y(dy)/(dx) = what the right side equals thru using the quotient rule?

I know the answer is dy/dx = 12/5y(4x + 3)^2 but I don't know how to get that

The Attempt at a Solution

 

[JonF]

How do you not know how to get it? You just said how! Use the quotient rule on the RHS.

 

[Caramon]

I'll explain, since apparently the person above me doesn't believe in clarifying?
[PLAIN]http://img101.imageshack.us/img101/3676/step1a.gif

Since this equation is comprised of simple polynomials you will be able to differentiate each term by itself.

Using the power rule and remembering how to apply implicit differentiation via. the Chain Rule:
[PLAIN]http://img585.imageshack.us/img585/1932/step2a.gif becomes [PLAIN]http://img87.imageshack.us/img87/9609/step2b.gif

To differentiate a single x with a constant in front, it just becomes the constant due to the power rule, so 4x becomes 4.

Instead of using the quotient rule for -3/4x I would put (4x)^-1 on top and multiply that by -3, so that you can use the power rule instead (which is 99% of the time much easier to deal with).
(-1)*(-3) = 3 and then reduce the power by 1 and you end up with (3(4x)^-2) which can be rewritten as (3)/(4x^2).

The derivative of any constant is zero, 3 is a constant therefore you can just ignore it.

So far we have:
[PLAIN]http://img59.imageshack.us/img59/8198/step3.gif

So we can just solve for y' by dividing both sides by 10y, if we decided to maybe add fractions on the other side then clean things up with some algebra we get:
[PLAIN]http://img41.imageshack.us/img41/6554/step4.gif

So... apparently the answer you "know" ... is wrong? Wolfram Alpha agrees with me:
http://www.wolframalpha.com/input/?i=derivative+of+5y^2+%3D+4x+-+3%2F4x+%2B+3

Hope this helped!

 

[JonF]

First you aren't supposed to give someone a complete answer. Second you entered it in wrong to wolfram, they most likely meant (4x – 3)/(4x + 3) not 4x – (3/4x) + 3. And what further clarification is needed? They stated exactly how to solve this problem and then said I don’t know what to do. If they got stuck on the quotient rule they need to show their work, not just be given an answer.

 

[Jan Hill]

How do you not know how to get it? You just said how! Use the quotient rule on the RHS.

You were right the 4x and 3 term are inside the brackets