What is implicit differentiation?

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Answer 1 · 2014-08-04T11:37:27

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

$d/dx(x^2)+d/dx(y^2)=d/dx(16)$

$2x+d/dx(y^2)=0$

To find $d/dx(y^2)$ we use the chain rule:

$d/dx=d/dy *dy/dx$

$d/dy(y^2)=2y*dy/dx$

$2x+2y*dy/dx=0$

Rearrange for $dy/dx$

$dy/dx=(-2x)/(2y$

$dy/dx=-x/y$

So essentially to use implicit differentiation you treat y the same as an x and when you differentiate it you multiply be $dy/dx$