Use implicit differentiation to find dy/dx for the equation x^3+y^3=3xy.

Makena Preston 2022-07-21 Answered
I have this practice problem before a test. Use implicit differentiation to find d y / d x for the equation
x 3 + y 3 = 3 x y .
I have no idea how to do this, I didn't understand my lecturer. Can you guys show me the steps?
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Answers (1)

Caylee Davenport
Answered 2022-07-22 Author has 14 answers
Think of y as a function of x. I will explicitly write this as y ( x ). Then proceed to differentiate everything with respect to x as normal, remembering the chain rule.

We need to differentiate x 3 + y 3 ( x ) = 3 x y ( x ). Let's do each term one by one.

1. Differentiate x 3 . You should quickly see this is 3 x 2 .
2. To differentiate ( y ( x ) ) 3 , we need to remember the chain rule. This can be written in many different ways, but this is a composition of the functions ( ) 3 y x. The derivative is 3 ( y ( x ) ) 2 y ( x )
3. To differentiate 3 x y ( x ), you must remember the product rule. The derivative is 3 y ( x ) + 3 x y ( x ).

So in total, differentiation yields
3 x 2 + 3 y 2 ( x ) y ( x ) = 3 y ( x ) + 3 x y ( x ) .
In this form, to find y , you isolate it (if possible) in this equation. Here, this simplifies (after cancelling factors of 3) to
y ( x ) = y ( x ) x 2 y 2 ( x ) x .

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0 = x 2 + y 2 1 y = 1 x 2
I differentiaded g ( x ) = 1 x 2 with the chain rule and got g ( x ) = x 1 x 2 .
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