How do you differentiate xcosy+ycos x=1

garnirativ8

garnirativ8

Answered question

2022-09-27

How do you differentiate x cos y + y cos x = 1

Answer & Explanation

cisalislewagobc

cisalislewagobc

Beginner2022-09-28Added 8 answers

Since we can't (or don't want to) solve for y, leave the function(s) implicit and use implicit differentiation. (An application of the chain rule.)
Given x cos y + y cos x = 1
We know d d x ( x cos y + y cos x ) = d d x ( 1 )
So, using the product rule twice on the left:
[ d x d x cos y + x d d x ( cos y ) ] + [ d y d x cos x + y d d x ( cos x ) ] = 0
Thus, we see that:
cos y - x sin y d y d x + d y d x cos x - y sin x = 0
Solve algebraically for d y d x to get:
d y d x = y sin x - cos y cos x - x sin y

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