I am trying to implicitly differentiate sin(x/y)=1/2

suchonosdy

suchonosdy

Answered question

2022-07-14

I am trying to implicitly differentiate
sin ( x / y ) = 1 / 2
The solution manual says
Step 1.
cos ( x / y ) y x d y d x y 2 = 0
But I don't understand how they arrive at this next part:
Step 2.
y x d y d x = 0
Is cos ( x / y ) = y 2 ?

Answer & Explanation

Alden Holder

Alden Holder

Beginner2022-07-15Added 15 answers

The idea is that when a b = 0, you have that a = 0 or b = 0 as solutions to the equation (or possibly both). Here, we have 1 y 2 cos ( x y ) = 0 or y x y = 0.

Since y cannot be "infinity", from the first solution we have that cos ( x y ) = 0 which says x y = π 2 , 3 π 2 , . So really, y is a linear function of x. But you can check that none of the above values I gave satisfy sin ( x y ) = 1 2 (in fact it is equal to ± 1 for these values). Check this yourself.

So that leaves us with y x y = 0. Is it clear now?
Greyson Landry

Greyson Landry

Beginner2022-07-16Added 5 answers

One approach says that the cosine cannot be 0 when the sine is 1/2, so if cos ( something ) ( something ) = 0, then the second "something" must be 0.
Another approach seems simpler, since it avoids differentiating any trigonometric functions and applying the chain rule to that differentiation. Just observe that
x y = arcsin 1 2 .
Differentiate the left side using the quotient rule and the chain rule. When you differentiate the right side, you get 0 since the right side is constant, and all references to trigonometric functions disappear.

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?