How do you find #(dy)/(dx)# given #-4x^2y^3+2=5x^2+y^2#?

Question

Luke Alvarez · Accepted Answer

#y'=(8xy^3+10x)/(-4x^2-3y^2-2y)#

Remember that Implicit Differentiation is really just a special case of the Chain Rule.

Every time that we differentiate the a factor or term what includes the variable y we have to include a factor of #dy/dx# or #y'#.

For the first term, #-4x^2y^3#, we have to use the Product Rule and Power Rule .
For the constant, #2#, we have to use the Constant Rule .
For the term, #5x^2#, use the Power Rule .
For the term, #y^2#, use the Power Rule .
#-4x^2 3y^2y'+(-8)xy^3+0=10x+2yy'#
Gather the terms with #y'# on one side of the equations and other terms on the other side.

#-4x^2 3y^2y'-2yy'=8xy^3+10x#
Factor out #y'#
#y'(-4x^2 3y^2-2y)=8xy^3+10x#
Isolate #y'# by dividing both sides by #(-4x^2 3y^2-2y)#
#y'cancel(-4x^2 3y^2-2y)/cancel(-4x^2 3y^2-2y)=(8xy^3+10x)/(-4x^2 3y^2-2y)#
#y'=(8xy^3+10x)/(-4x^2 3y^2-2y)#
I have a couple of tutorials on Implicit Differentiation here, https://tutor.hix.ai

Christopher Agresta · Answer

undefined To find $ \frac{{dy}}{{dx}} $ given the equation $ -4x^2y^3 + 2 = 5x^2 + y^2 $, differentiate both sides of the equation implicitly with respect to $ x $. Then solve for $ \frac{{dy}}{{dx}} $.