whichagreeswith the previouscalculations.This secondmethodis calledtheimplicitdifferentiation
method.You may wonderand say that the first methodis easierand fasterand thereis no reasonfor the
secondmethod.That’s probablytrue, but considerthis function:
How wouldyou solvefory? That wouldbe a difficult task. So the methodof implicitdifferentiationsometimes
is very useful,especiallywhenit is inconvenientor impossibleto solveforyin termsofx. Explicitlydefined
functionsmay be writtenwith a directrelationshipbetweentwo variableswith clearindependentand depen-
dent variables.Implicitlydefinedfunctionsor relationsconnectthe variablesin a way that makesit impossible
to separatethe variablesinto a simpleinputoutputrelationship.Morenoteson explicitand implicitfunctions
can be foundat http://en.wikipedia.org/wiki/Implicit_function.
Example1:
Finddy/dxif
Solution:
Differentiatingboth sideswith respecttoxand then solvingfordy/dx,
Solvingfordy/dx, we finallyobtain
Implicitdifferentiationcan be usedto calculatethe slopeof the tangentline as the examplebelowshows.
Example2:
Find the equationof the tangentline that passesthroughpoint(1, 2) to the graphof
Solution: