How does implicit differentiation work?

odigavz

odigavz

Open question

2022-08-18

How does implicit differentiation work?

Answer & Explanation

alienceenvedsf0

alienceenvedsf0

Beginner2022-08-19Added 15 answers

Implicit differentiation is a way of differentiating when you have a function in terms of both x and y. For example:
x 2 + y 2 = 16
This is the formula for a circle with a centre at (0,0) and a radius of 4
So using normal differentiation rules x 2 and 16 are differentiable if we are differentiating with respect to x
d d x ( x 2 ) + d d x ( y 2 ) = d d x ( 16 )
2 x + d d x ( y 2 ) = 0
To find d d x ( y 2 ) we use the chain rule:
d d x = d d y d y d x
d d y ( y 2 ) = 2 y d y d x
2 x + 2 y d y d x = 0
Rearrange for d y d x
d y d x = - 2 x 2 y
d y d x = - x y
So essentially to use implicit differentiation you treat y the same as an x and when you differentiate it you multiply be d y d x

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