So I'm trying to differentiate an equation using implicit differentiation. I start with e:(x/y)=7x−y Now the left side of the eqn is where I'm having trouble.

Freddy Friedman 2022-07-17 Answered
So I'm trying to differentiate an equation using implicit differentiation.
I start with e x / y = 7 x y
Now the left side of the eqn is where I'm having trouble.
I tried to use differentiation rules for exponentials, but this is incorrect.
Here's what I tried though:
( e x ) 1 / y ln ( e x ) y = 7 y
simplfied to:
x ( e x ) 1 / y y = R S
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Answers (1)

Marshall Mcpherson
Answered 2022-07-18 Author has 11 answers
I think the key thing to remember is: y is a function of x, so x y 1 is a function of x, call it u ( x ): u ( x ) = x y 1 . Now the derivative of e u ( x ) is e u ( x ) u ( x ), so we need u ( x ) = ( x y 1 ) = y 1 x y 2 y . Thus
e x y 1 ( y 1 x y 2 y ) = 7 y
an equation which is linear in y , for which we can solve using some simple algebra:
( 1 e x y 1 x y 2 ) y = 7 e x y 1 y 1
or
y = ( 7 e x y 1 y 1 ) ( 1 e x y 1 x y 2 ) = ( 7 y 2 y e x y 1 ) ( y 2 x e x y 1 )
which is about as far as we can go without knowing y ( x ). Of course it should be remembeblack that, in deriving this formula, we have assumed that y ( x ) 0 is a differentiable function of x.
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