Understanding the Two Solutions for Derivatives of (2x+y)/(x-3y)=4

[Inspector Gadget]

This was given as a problem...

$$\frac{2x+y}{x-3y}=4$$

Now, if you just solve it as is, using implicit differentiation, you get the deriviative to be y/x.

However, if you multiply by the denominator, you get...

$$2x+y = 4(x-3y)$$

...or...

$$2x+y=4x-12y$$

...and if you do implicit differentiation, you get the deriviate to be 2/13.

Why are there two solutions like that, and is any of the two any more right than the other?

 

[matt grime]

since y = 2x/13, I'm not sure i see what your problem is.

 

[jdavel]

Inspector Gadget asked: "Why are there two solutions like that, and is any of the two any more right than the other?"

Yes, the correct answer is 2/13. You're trying to find y', which is a function of x. So when you get to the solution y' = y/x, you need to solve for y as a function of x to complete your solution for y'. As Matt Grime showed you, when you plug y(x) into y'(x) you get 2/13.

In this case it would seem that implicit differentiation doesn't do you much good. Might as well just solve the orignal algebraic equation for y and then differentiate explicitly.

 

[Inspector Gadget]

Yeah, that's basically what I was asking.

We were tested on implicit differentiation last week, and I simply did the second method by multiplying by the denominator instead of having to do the quotient rule, then FOIL twice, simplify, etc., and got 2/13. It was marked wrong. I asked why and showed all my work to prove that you get 2/13, man at the helm said just said it wasn't correct. I honestly didn't even realize that the equation gives you a straight line at the time when I was arguing if you solve for y, so that should make it even easier to get my point across now.

 

[matt grime]

As you were tested on implicit differentiation, then the question presumably meant you to use implicit differentiation, and not rearrangement, so I can see why the answer was marked 'wrong', even if I don't agree with the 'man at the helm's' response when he didn't explain why it was wrong. If the question asks you to solve using a specific method and does not explicitly state that you are allowed to use other methods, then you must use the method specified, irrespective of if the other method is simpler. If you wish to use another method you must prove that it produces the same answer, which is quite a big task, and you didn't state as that y=2x/13 the answers are equivalent.

 

[Inspector Gadget]

I'll have to see what exactly the directions said, but his response seemed to come off as "the deriviative you supplied isn't a deriviative of the equation" than "you didn't follow the directions".

And usually, if you don't follow directions, you get half credit if you get the correct answer.

I'll definitely seem him for class by Thursday...I'll have to nag him about it again.

 

[matt grime]

Well different people, different rules and all that - if you did that [didn't use the method as instructed] at my university, or even at high school, you got no marks.