Interpretation of Trig Expression

[skullers_ab]

[SOLVED] Interpretation of Trig Expression

Homework Statement

Question: Sin xy = y, find dy/dx. This was given as Homework.
I did the question as Sin (xy) = y, whereas my teacher took the question as (Sin x)y = y and made a joke out of me in the class for thinking otherwise. All I want to know is which way should the question be interpreted.

Homework Equations

Sin xy = y

The Attempt at a Solution

Taking my interpretation of Sin xy = y as Sin (xy) = y, the differentiated equation I get is:
dy/dx = (-y Cos xy)/(x Cos xy - 1)

 

[konthelion]

Hello.

I believe you have to use implicit differentiation. Equate the derivatives of both sides
$$\frac{dy}{dx} sin(xy) = \frac{dy}{dx}(y)$$. Remember, y is not a constant, but a differentiable function of x.

Therefore,
$$cos(xy)(y+x\frac{dy}{dx})=dy/dx$$
You can continue from here, to solve for $$dy/dx$$

 

[tiny-tim]

Welcome to PF!

Question: Sin xy = y, find dy/dx.
…
All I want to know is which way should the question be interpreted.

Hi skullers_ab! Welcome to PF!

(Does this come from a book? If your teacher made it up, then it means whatever the teacher says it means!)

Well, I'm inclined to agree with you …

sin(xy) = y is a perfectly sensible equation …

and it's a pretty pointless exercise if it's just (sinx)y = y!

 

[skullers_ab]

@konthelion
Yes, I am aware of the steps/method involved to solve the question, it's just that I needed to know the correct interpretation of the question. Thank you for your response anyways.

@tiny-tim
Thank you for your favourable response.

This question is from a past year's external exam question paper, Fiji Seventh Form Examination - an exam done nationwide by the last year of High School students in Fiji. Therefore it is not 'made up' by the teacher.

I tried to explain that to the teacher as well, that it would be pointless if the question was otherwise. His EGO gets in the way of his logic all the time. I could get started on numerous similar situations where, I swear, he would've killed me, if he had a gun, for stating:
1. Opposite of what he said
2. The right thing (As confirmed by textbooks and internet research).

Just a few I can remember off the top of my head:

1. He was arguing with me and told me that it IS possible to solve for 3 variables with only two simultaneous equations using concepts involving advanced matrices. I said that according to my understanding it is NOT possible to have only one set of values for the three variables with only two equations. Basically they can't be solved to get a single answer.

2. He argued that anything raised to a very high power is approximately equal to zero. This is when we were discussing finding the sum to infinity under geometric progression (series). In fact it is (1/k)^(a very large no.) is approx = 0, not k^(a very large no.) = 0.

3. In trying to ask him to explain to me the concept of Radioactive decay, he concluded that it has nothing to do with half life. "An element does not go through Beta decay, it is PUT under beta decay and that's when beta decay happens".

4. He said differentiating the expression (1/y) should be done using the quotient rule and differentiating it by re-writing it as (y^-1) and then using simple differentiation rules to differentiate it is WRONG, even though the answer you get is the same.

5. etcetera

Please do let me know if he was right in anyone of the above situations.
Thank you for confirming.

*Sorry for making this post more of a personal vendetta seeming outburst rather than a math help topic.

 

[rock.freak667]

1. He was arguing with me and told me that it IS possible to solve for 3 variables with only two simultaneous equations using concepts involving advanced matrices. I said that according to my understanding it is NOT possible to have only one set of values for the three variables with only two equations. Basically they can't be solved to get a single answer.

Not too educated on linear algebra but solving two simultaneous equations with three unknowns would give you the straight line in which the two planes intersect i.e. an infinite number of solutions

2. He argued that anything raised to a very high power is approximately equal to zero. This is when we were discussing finding the sum to infinity under geometric progression (series). In fact it is (1/k)^(a very large no.) is approx = 0, not k^(a very large no.) = 0.

Anything raised to a high power is not always approximately zero. take any integer and raise it to the power of 1000, clearly it isn't zero. But

$$\lim_{n \rightarrow \infty} \frac{a}{x^n}=0$$

where a is a constant

3. In trying to ask him to explain to me the concept of Radioactive decay, he concluded that it has nothing to do with half life. "An element does not go through Beta decay, it is PUT under beta decay and that's when beta decay happens".

Not too sure on this one, as I think elements go through it.

4. He said differentiating the expression (1/y) should be done using the quotient rule and differentiating it by re-writing it as (y^-1) and then using simple differentiation rules to differentiate it is WRONG, even though the answer you get is the same.

Differentiating it by the quotient rule, will give the same as if you write it as an exponent like that.

 

[skullers_ab]

Thanks, my doubts are cleared. And as for my initial query of Sin xy = 0, that is confirmed to be taken the way I took it, not otherwise.

I have used PF for the first time for help, and I love it.

Thank you all for replying.