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3.3: Implicit Differentiation

Created by: CK-12

This activity is intended to supplement Calculus, Chapter 2, Lesson 6.

Problem 1 – Finding the Derivative of x^2 + y^2 = 36

The relation, x^2 + y^2 = 36, in its current form implicitly defines two functions, f_1(x) = y and  f_2(x) = y. Find these two functions by solving x^2 + y^2 = 36 for y.

f_1(x) = && f_2(x) =

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

x^2 + (f_1(x))^2 = 36 && x^2 + (f_2(x))^2 = 36

This confirms that f_1(x) and f_2(x) explicitly defines the relation x^2 + y^2 = 36.

Graph f_1(x) and f_2(x) 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 x = 2.

  • 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 (x,y) located on the curve is by taking the derivative of f_1(x) and f_2(x).

\frac{dy}{dx}f_1(x)= && \frac{dy}{dx}f_2(x)=

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

Substitute 2 for x to determine the slope of the tangents to x^2 + y^2 = 36 at x = 2. \frac{dy}{dx}f_1(2)= && \frac{dy}{dx}f_2(2)=

Another way to find the slope of a tangent is by finding the derivative of x^2 + y^2 = 36 using implicit differentiation. On the Calculator screen press F3:Calc > D:impDif( to access the impDif command. Enter impDif (x^2 + y^2 = 36,x, y) to find the derivative.

\frac{dy}{dx}=

Use this result to find the slope of the tangents to x^2 + y^2 = 36 at x = 2. First you will need to find the y-values when x = 2.

\frac{dy}{dx}(2,y)= && \frac{dy}{dx}(2,y)=

  • Is your answer consistent with what was found earlier?
  • Rewrite the implicit differentiation derivative in terms of x. Show that, for all values of x and y, the derivatives of f_1(x) and f_2(x) that you found earlier are equal to the result found using the impDif command.

Problem 2 – Finding the Derivative of x^2 + y^2 = 36 by Hand

To find the derivative of a relation F(x, y), take the derivative of y with respect to x of each side of the relation. Looking at the original example, x^2 + y^2 = 36, we get:

\frac{d}{dx}(x^2 + y^2) &= \frac{d}{dx}(36)\\\frac{d}{dx}(x^2) + \frac{d}{dx}(y^2) &= \frac{d}{dx}(36)

Evaluate the following and by hand.

\frac{d}{dx}(x^2) = && \frac{d}{dx}(36)=

Use the Derivative command to find \frac{d}{dx}(y^2). Set up the expression up as \frac{d}{dx}(y(x)^2) . Notice that y(x) is used rather than just y. This is very important because it reminds the calculator that y is a function of x.

\frac{d}{dx}(y^2) =

You have now evaluated \frac{d}{dx}(x^2),\frac{d}{dx}(y^2), and \frac{d}{dx}(36). Replace these expressions in the equation \frac{d}{dx}(x^2) + \frac{d}{dx}(y^2) = \frac{d}{dx}(36) and solve for \frac{dy}{dx}.

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

Problem 3 – Finding the Derivative of y^2 + xy = 2

The relation, y^2 + xy = 2, can also be solved as two functions, f_1(x) and f_2(x), which explicitly define it.

  • What strategy can be used to solve y^2 + xy = 2 for y?

Solve y^2 + xy = 2 for y and use the Solve command (press F2:Algebra > 1:solve() to check your answer.

The derivative of y^2 + xy = 2 can then be found by taking the derivatives of f_1(x) and f_2(x). However, the derivative can be found more easily using implicit differentiation.

Use implicit differentiation to find the derivative of y^2 + xy = 2. Check your result by using the impDif command. (Hint: The product rule must be used to find the derivative of xy.)

 \frac{dy}{dx}=

Use the derivative you found for y^2 + xy = 2 to calculate the slope at x = -6. First you will need to find the y-values when x = -6.

 \frac{dy}{dx}(-6,y)= && \frac{dy}{dx}(-6,y)=

Verify your result graphically. Graph the two functions, f_1(x) and f_2(x). Then use the slopes and points to graph each tangent line.

Extension – Finding the Derivative of x^3 + y^3 = 6xy

The relation x^3 + y^3 = 6xy cannot be solved explicitly for y. In this case implicit differentiation must be used.

  • Find the derivative of x^3 + y^3 = 6xy and use the impDif command to verify your result.

 \frac{dy}{dx}=

Use this result to find the slope of the tangents to x^3 + y^3 = 6xy at x = 1. (Hint: Use the solve command to find the y values that correspond to x = 1.)

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Feb 23, 2012

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Aug 19, 2014
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TI.MAT.ENG.SE.1.Calculus.3.3

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