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Implicit Differentiation- DS5

Objectives

  • Find the derivative of a complicated function by using implicit differentiation.
  • Use implicit differentiation to determine the equation of a tangent line.

Summary

In an equation involving x and y where portions of the graph can be defined by explicit functions of x , we say that y is an implicit function of x . A good example of such a curve is the unit circle.

We use implicit differentiation to differentiate an implicitly defined function. We differentiate both sides of the equation with respect to x , treating y as a function of x by applying the chain rule. If possible, we subsequently solve for d y d x using algebra.

While d y d x may now involve both the variables x and y , d y d x still gives the slope of the tangent line to the curve. It may be used to decide where the tangent line is horizontal d y d x = 0 or vertical ( d y d x is undefined), or to find the equation of the tangent line at a particular point on the curve.

See the Desmos Demonstration.

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