How does implicit differentiation work?

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Answer 1 · 2014-08-05T09:41:47

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#

Theres another video on the subject here