I was just wondering where the y'/(dy/dx) in implicit differentiation comes from. x^2+y^2=25 (d/dx)x^2+(d/dy)y^2∗∗(dy/dx)∗∗=25(d/dx) 2x+2y(dy/dx)=0 (dy/dx)=−x/y Where does the bold part come from? Wikipedia says it's a byproduct of the chain rule, but it's just not clicking for me.

babuliaam 2022-09-26 Answered
I was just wondering where the y'/(dy/dx) in implicit differentiation comes from
x 2 + y 2 = 25
( d / d x ) x 2 + ( d / d y ) y 2 ( d y / d x ) = 25 ( d / d x )
2 x + 2 y ( d y / d x ) = 0
( d y / d x ) = x / y
Where does the bold part come from?
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Answers (2)

Simeon Hester
Answered 2022-09-27 Author has 16 answers
When you implicitly differentiate x 2 + y 2 = 25, you are differentiating with respect to a particular variable—in this case, x, so:
d d x ( x 2 + y 2 ) = d d x 25 d d x ( x 2 ) + d d x ( y 2 ) = 0 2 x + 2 y d y d x = 0 2 y d y d x = 2 x d y d x = x y
From the 3rd line to the 4th line, d d x ( y 2 ) is the derivative with respect to x of y 2 , in which (as in Ryan Budney's comment) we assume that y is some function of x, so we apply the chain rule, differentiating y 2 with respect to y and multiplying by the derivative of y with respect to x to get 2 y d y d x .


edit: I think it might be useful if I introduced a slightly different notation: Let D x be the differential operator with respect to x, which you have previously written as d d x (and, similarly, D y is the differential operator with respect to y). When we apply the differential operator to something, we read and write it like a function: D x ( x 2 ) = 2 x is "the derivative with respect to x of x 2 is 2 x."

Now, rewriting the work above in this notation:
D x ( x 2 + y 2 ) = D x ( 25 ) D x ( x 2 ) + D x ( y 2 ) = 0 2 x + D y ( y 2 ) D x ( y ) = 0 2 x + 2 y D x ( y ) = 0 2 y D x ( y ) = 2 x D x ( y ) = d y d x = x y
And, to your question of finding d x d y :
D y ( x 2 + y 2 ) = D y ( 25 ) D y ( x 2 ) + D y ( y 2 ) = 0 D x ( x 2 ) D y ( x ) + 2 y = 0 2 x D y ( x ) + 2 y = 0 2 x D y ( x ) = 2 y D y ( x ) = d x d y = y x
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saucletbh
Answered 2022-09-28 Author has 3 answers
In symbols, the chain rule gives:
d ( y 2 ) d x = d ( y 2 ) d y d y d x = 2 y d y d x
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