How does implicit differentiation work?

Question

Emilia Aguilar · Accepted Answer

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#

Youtube Implicit Differentiation

Theres another video on the subject here

Samantha Adkison · Answer

undefined Implicit differentiation is a technique used to find the derivative of functions that are not explicitly expressed in terms of one variable. It involves differentiating both sides of an equation with respect to the variable of interest, treating the dependent variable as a function of the independent variable. This allows for the calculation of derivatives of functions where the dependent variable is not isolated explicitly.