How to find dy/dx by implicit differentiation given x^2+3xy+y^2=0?

mriswith170wp

mriswith170wp

Answered question

2023-02-13

How to find d y d x by implicit differentiation given x 2 + 3 x y + y 2 = 0 ?

Answer & Explanation

leoanardig7k

leoanardig7k

Beginner2023-02-14Added 7 answers

Given: x 2 + 3 x y + y 2 = 0
Distinguish each term in relation to x:
d ( x 2 ) d x + 3 d ( x y ) d x + d ( y 2 ) d x = d ( 0 ) d x
Use the power rule, d y d x = n x x - 1 , on the first term:
2 x + 3 d ( x y ) d x + d ( y 2 ) d x = d ( 0 ) d x
Use the product rule, d ( x y ) d x = d x d x y + x d y d x = y + x d y d x on the second term:
2 x + 3 ( y + x d y d x ) + d ( y 2 ) d x = d ( 0 ) d x
Use the chain rule, d ( y 2 ) d x = 2 y d y d x , on the third term:
2 x + 3 ( y + x d y d x ) + 2 y d y d x = d ( 0 ) d x
The derivative of a constant is 0:
2 x + 3 ( y + x d y d x ) + 2 y d y d x = 0
Distribute the 3:
2 x + 3 y + 3 x d y d x + 2 y d y d x = 0
Move all of the terms that do not contain d y d x to the right:
3 x d y d x + 2 y d y d x = - ( 2 x + 3 y )
Factor out d y d x :
( 3 x + 2 y ) d y d x = - ( 2 x + 3 y )
Divide by 3 x + 2 y :
d y d x = - 2 x + 3 y 3 x + 2 y
leshumict6

leshumict6

Beginner2023-02-15Added 2 answers

differentiate implicitly with respect to x
the term 3 x y is differentiated using the product rule
2 x + 3 ( x . d y d x + y .1 ) + 2 y . d y d x = 0
2 x + 3 x d y d x + 3 y + 2 y d y d x = 0
d y d x ( 3 x + 2 y ) = - 2 x - 3 y
d y d x = - 2 x + 3 y 3 x + 2 y

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