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3.3: Implicit Differentiation

Difficulty Level: At Grade Created by: CK-12
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This activity is intended to supplement Calculus, Chapter 2, Lesson 6.

Problem 1 – Finding the Derivative of

The relation, , in its current form implicitly defines two functions, and . Find these two functions by solving for .

Substitute the above functions in the original relation and then simplify.

This confirms that and explicitly defines the relation .

Graph and on the same set of axis and then draw it in the space to the right. Imagine that you were asked to find the slope of the curve at .

  • Why might this question be potentially difficult to answer?
  • What strategies or methods could you use to answer this question?

One way to find the slope of a tangent drawn to the circle at any point located on the curve is by taking the derivative of and .

Check that your derivatives are correct by using the Derivative command (press F3:Calc > 1:d( differentiate) on the Calculator screen.

Substitute for to determine the slope of the tangents to at .

Another way to find the slope of a tangent is by finding the derivative of using implicit differentiation. On the Calculator screen press F3:Calc > D:impDif( to access the impDif command. Enter impDif to find the derivative.

Use this result to find the slope of the tangents to at . First you will need to find the values when .

  • Is your answer consistent with what was found earlier?
  • Rewrite the implicit differentiation derivative in terms of . Show that, for all values of and , the derivatives of and that you found earlier are equal to the result found using the impDif command.

Problem 2 – Finding the Derivative of by Hand

To find the derivative of a relation , take the derivative of with respect to of each side of the relation. Looking at the original example, , we get:

Evaluate the following and by hand.

Use the Derivative command to find . Set up the expression up as . Notice that is used rather than just . This is very important because it reminds the calculator that is a function of .

You have now evaluated , and . Replace these expressions in the equation and solve for .

Compare your result to the one obtained using the impDif command.

Problem 3 – Finding the Derivative of

The relation, , can also be solved as two functions, and , which explicitly define it.

  • What strategy can be used to solve for ?

Solve for and use the Solve command (press F2:Algebra > 1:solve() to check your answer.

The derivative of can then be found by taking the derivatives of and . However, the derivative can be found more easily using implicit differentiation.

Use implicit differentiation to find the derivative of . Check your result by using the impDif command. (Hint: The product rule must be used to find the derivative of .)

Use the derivative you found for to calculate the slope at . First you will need to find the values when .

Verify your result graphically. Graph the two functions, and . Then use the slopes and points to graph each tangent line.

Extension – Finding the Derivative of

The relation cannot be solved explicitly for . In this case implicit differentiation must be used.

  • Find the derivative of and use the impDif command to verify your result.

Use this result to find the slope of the tangents to at . (Hint: Use the solve command to find the values that correspond to .)

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