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

babuliaam

Answered question

2022-09-26

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?

Answer & Explanation

Simeon Hester

Simeon Hester

Beginner2022-09-27Added 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
saucletbh

saucletbh

Beginner2022-09-28Added 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

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?