How to find dy/dx from x^3y^6=(x+y)^9 using implicit differentiation?

2k1ablakrh0 2022-09-27 Answered
How to find
d y / d x
from
x 3 y 6 = ( x + y ) 9
using implicit differentiation? I tried solving but I ended up with solution that does not agree with my textbook answer. How can I get
d y / d x = y / x
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Answers (2)

rmercierm7
Answered 2022-09-28 Author has 4 answers
The answer does appear to be correct. Here's how to go about it:

1) First, use implicit differentiation to get
3 x 2 y 6 + 6 x 3 y 5 d y d x = 9 ( x + y ) 8 ( 1 + d y d x ) .
2) Solve for d y d x to get
d y d x = 9 ( x + y ) 8 3 x 2 y 6 6 x 3 y 5 9 ( x + y ) 8 .
3) Multiply the numerator and denominator of the RHS by ( x + y ) to get
d y d x = 9 ( x + y ) 9 3 x 2 y 6 ( x + y ) 6 x 3 y 5 ( x + y ) 9 ( x + y ) 9 .
4) Using the original formula for ( x + y ) 9 , this becomes
d y d x = 9 x 3 y 6 3 x 2 y 6 ( x + y ) 6 x 3 y 5 ( x + y ) 9 x 3 y 6 .
5) By simplifying, we get
d y d x = 6 x 3 y 6 3 x 2 y 7 6 x 4 y 5 3 x 3 y 6 = 2 x y y 2 2 x 2 x y .
6) By factoring common factors, we get
d y d x = y ( 2 x y ) x ( 2 x y ) = y x .
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babuliaam
Answered 2022-09-29 Author has 1 answers
First simplify your equation by taking the cube root of each side to get
x y 2 = ( x + y ) 3
Then take the derivative of both sides and do the usual process to find
d y d x = 3 ( x + y ) 2 y 2 2 x y 3 ( x + y ) 2
Then multiplying numerator and denominator by x + y,
d y d x = 3 ( x + y ) 3 y 2 ( x + y ) 2 x y ( x + y ) 3 ( x + y ) 2
= 3 x y 2 x y 2 y 3 2 x 2 y + 2 x y 2 3 x y 2
= 2 x y y 3 2 x 2 y x y 2
= y 2 ( 2 y ) x y ( 2 y )
= y x
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