use implicit differentiation to find d y d x in terms of x and y1.)...

smuklica8i

smuklica8i

Answered

2022-07-22

use implicit differentiation to find d y d x in terms of x and y

1.) x 3 x y + y 2 = 4

2.) y = sin ( x y )
find d 2 y d x 2

3.) x 2 y 2 2 x = 3

Answer & Explanation

Jazlene Dickson

Jazlene Dickson

Expert

2022-07-23Added 15 answers

I'll show you how to do 1), and you can try the others yourself. Differentiating implicitly, we have
d d x ( x 3 x y + y 2 ) = d d x 4
or
3 x 2 y x y + 2 y y = 0
so that
3 x 2 y = y ( x 2 y )
and hence
y = 3 x 2 y x 2 y .
Livia Cardenas

Livia Cardenas

Expert

2022-07-24Added 5 answers

And 2,
d y = cos ( x y ) ( y d x + x d y ) = y cos ( x y ) d x + x cos ( x y ) d y
using chain and product rules. Then solve for d y, and then finally d y d x ,
d y x cos ( x y ) d y = cos ( x y ) ( y d x + x d y ) = y cos ( x y ) d x d y ( 1 x cos ( x y ) ) = y cos ( x y ) d x d y d x = y cos ( x y ) 1 x cos ( x y )

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