I have the following expression which I need to implicitly differentiate: xy^2+x^2+y+sin(x^2y)=0

Jean Farrell 2022-09-24 Answered
I have the following expression which I need to implicitly differentiate:
x y 2 + x 2 + y + sin ( x 2 y ) = 0
I'm a little confused as I'm not entirely sure what to do with the trig function. Here is my work so far:
d y d x [ x y 2 + x 2 + y + sin ( x 2 y ) ] = d y d x 0
d y 2 d x + 2 x + d y d x + cos ( x 2 y ) ( 2 x d y d x ) = 0
How should I proceed?
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Answers (2)

Simeon Hester
Answered 2022-09-25 Author has 16 answers
d d x [ x y 2 + x 2 + y + sin ( x 2 y ) ] = d d x ( 0 ) y 2 + 2 x y d y d x + 2 x + d y d x + cos ( x 2 y ) ( 2 x y + x 2 d y d x ) = 0
We use the product rule and chain rule here, and also the operator for differentiation is
d d x
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zaiskaladu
Answered 2022-09-26 Author has 2 answers
First, you should take the derivative of both sides of the equation (apply d d x instead of d y d x ). Then make use of the chain and product rules. It helps to think of y as a function of x (i.e. y = y ( x )).
d d x [ x y 2 + x 2 + y + s i n ( x 2 y ) ] = d d x 0 y 2 + x ( 2 y ) d y d x + 2 x + d y d x + cos ( x 2 y ) [ d d x ( x 2 y ) ] = 0 y 2 + x ( 2 y ) d y d x + 2 x + d y d x + cos ( x 2 y ) ( 2 x y + x 2 d y d x ) = 0 d y d x ( 2 x y + 1 + x 2 cos ( x 2 y ) ) = ( y 2 + 2 x ) d y d x = ( y 2 + 2 x ) ( 2 x y + 1 + x 2 cos ( x 2 y ) )
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