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Learning Objectives

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

We have already studied how to find equations of tangent lines to functions and the rate of change of a function at a specific point. In all these cases we had the explicit equation for the function and differentiated these functions explicitly. Suppose instead that we want to determine the equation of a tangent line to an arbitrary curve or the rate of change of an arbitrary curve at a point. In this section, we solve these problems by finding the derivatives of functions that define y implicitly in terms of x.

Implicit Differentiation

In most discussions of math, if the dependent variable y is a function of the independent variable x, we express y in terms of x. If this is the case, we say that y is an explicit function of x. For example, when we write the equation y=x2+1, we are defining y explicitly in terms of x. On the other hand, if the relationship between the function y and the variable x is expressed by an equation where y is not expressed entirely in terms of x, we say that the equation defines y implicitly in terms of x. For example, the equation yx2=1 defines the function y=x2+1 implicitly.

Implicit differentiation allows us to find slopes of tangents to curves that are clearly not functions (they fail the vertical line test). We are using the idea that portions of y are functions that satisfy the given equation, but that y is not actually a function of x.

In general, an equation defines a function implicitly if the function satisfies that equation. An equation may define many different functions implicitly. For example, the functions

y=25x2, y=25x2, and y={25x2if5x<025x2if0x5, which are illustrated in (Figure) , are just three of the many functions defined implicitly by the equation x2+y2=25.

The circle with radius 5 and center at the origin is graphed fully in one picture. Then, only its segments in quadrants I and II are graphed. Then, only its segments in quadrants III and IV are graphed. Lastly, only its segments in quadrants II and IV are graphed.
Figure 1. The equation x2+y2=25 defines many functions implicitly.

If we want to find the slope of the line tangent to the graph of x2+y2=25 at the point (3,4), we could evaluate the derivative of the function y=25x2 at x=3. On the other hand, if we want the slope of the tangent line at the point (3,4), we could use the derivative of y=25x2. However, it is not always easy to solve for a function defined implicitly by an equation. Fortunately, the technique of implicit differentiation allows us to find the derivative of an implicitly defined function without ever solving for the function explicitly. The process of finding dydx using implicit differentiation is described in the following problem-solving strategy.

Problem-Solving Strategy: Implicit Differentiation

To perform implicit differentiation on an equation that defines a function y implicitly in terms of a variable x, use the following steps:

  1. Take the derivative of both sides of the equation. Keep in mind that y is a function of x. Consequently, whereas ddx(sinx)=cosx,ddx(siny)=cosydydx because we must use the Chain Rule to differentiate siny with respect to x.
  2. Rewrite the equation so that all terms containing dydx are on the left and all terms that do not contain dydx are on the right.
  3. Factor out dydx on the left.
  4. Solve for dydx by dividing both sides of the equation by an appropriate algebraic expression.

Using Implicit Differentiation

Assuming that y is defined implicitly by the equation x2+y2=25, find dydx.

Solution

Follow the steps in the problem-solving strategy.

ddx(x2+y2)=ddx(25)Step 1. Differentiate both sides of the equation.ddx(x2)+ddx(y2)=0Step 1.1. Use the sum rule on the left.On the right,ddx(25)=0.2x+2ydydx=0Step 1.2. Take the derivatives, soddx(x2)=2xandddx(y2)=2ydydx.2ydydx=2xStep 2. Keep the terms withdydxon the left.Move the remaining terms to the right.dydx=xyStep 4. Divide both sides of the equation by2y.(Step 3 does not apply in this case.)

Analysis

Note that the resulting expression for dydx is in terms of both the independent variable x and the dependent variable y. Although in some cases it may be possible to express dydx in terms of x only, it is generally not possible to do so.

Using Implicit Differentiation and the Product Rule

Assuming that y is defined implicitly by the equation x3siny+y=4x+3, find dydx.

Solution

ddx(x3siny+y)=ddx(4x+3)Step 1: Differentiate both sides of the equation.ddx(x3siny)+ddx(y)=4Step 1.1: Apply the sum rule on the left.On the right,ddx(4x+3)=4.(ddx(x3)siny+ddx(siny)x3)+dydx=4Step 1.2: Use the product rule to findddx(x3siny).Observe thatddx(y)=dydx.3x2siny+(cosydydx)x3+dydx=4Step 1.3: We knowddx(x3)=3x2.Use thechain rule to obtainddx(siny)=cosydydx.x3cosydydx+dydx=43x2sinyStep 2: Keep all terms containingdydxon theleft. Move all other terms to the right.dydx(x3cosy+1)=43x2sinyStep 3: Factor outdydxon the left.dydx=43x2sinyx3cosy+1Step 4: Solve fordydxby dividing both sides ofthe equation byx3cosy+1.

Using Implicit Differentiation to Find a Second Derivative

Find d2ydx2 if x2+y2=25.

Solution

In (Figure) , we showed that dydx=xy. We can take the derivative of both sides of this equation to find d2ydx2.

d2ydx2=ddx(xy)Differentiate both sides ofdydx=xy.=(1yxdydx)y2Use the quotient rule to findddx(xy).=y+xdydxy2Simplify.=y+x(xy)y2Substitutedydx=xy.=y2x2y3Simplify.

At this point we have found an expression for d2ydx2. If we choose, we can simplify the expression further by recalling that x2+y2=25 and making this substitution in the numerator to obtain d2ydx2=25y3.

Find dydx for y defined implicitly by the equation 4x5+tany=y2+5x.

Hint

Follow the problem solving strategy, remembering to apply the chain rule to differentiate tany and y2.

Solution

dydx=520x4sec2y2y

Finding Tangent Lines Implicitly

Now that we have seen the technique of implicit differentiation, we can apply it to the problem of finding equations of tangent lines to curves described by equations.

Finding a Tangent Line to a Circle

Find the equation of the line tangent to the curve x2+y2=25 at the point (3,4).

Solution

Although we could find this equation without using implicit differentiation, using that method makes it much easier. In (Figure) , we found dydx=xy.

The slope of the tangent line is found by substituting (3,4) into this expression. Consequently, the slope of the tangent line is dydx|(3,4)=34=34.

Using the point (3,4) and the slope 34 in the point-slope equation of the line, we then solve for y to obtain the equation y=34x254 ( (Figure) ).

The circle with radius 5 and center at the origin is graphed. A tangent line is drawn through the point (3, -4).
Figure 2. The line y=34x254 is tangent to x2+y2=25 at the point (3,4).

Finding the Equation of the Tangent Line to a Curve

Find the equation of the line tangent to the graph of y3+x33xy=0 at the point (32,32) ( (Figure) ). This curve is known as the folium (or leaf) of Descartes.

A folium is shown, which is a line that creates a loop that crosses over itself. In this graph, it crosses over itself at (0, 0). Its tangent line from (3/2, 3/2) is shown.
Figure 3. Finding the tangent line to the folium of Descartes at (32,32).

Solution

Begin by finding dydx.

ddx(y3+x33xy)=ddx(0)3y2dydx+3x2(3y+3xdydx)=0dydx=3y3x23y23x.

Next, substitute (32,32) into dydx=3y3x23y23x to find the slope of the tangent line:

dydx|(32,32)=1.

Finally, substitute into the point-slope equation of the line and solve for y to obtain

y=x+3.

Applying Implicit Differentiation

In a simple video game, a rocket travels in an elliptical orbit whose path is described by the equation 4x2+25y2=100. The rocket can fire missiles along lines tangent to its path. The object of the game is to destroy an incoming asteroid traveling along the positive x-axis toward (0,0). If the rocket fires a missile when it is located at (3,83), where will it intersect the x-axis?

Solution

To solve this problem, we must determine where the line tangent to the graph of

4x2+25y2=100 at (3,83) intersects the x-axis. Begin by finding dydx implicitly.

Differentiating, we have

8x+50ydydx=0.

Solving for dydx, we have

dydx=4x25y.

The slope of the tangent line is dydx|(3,83)=950. The equation of the tangent line is y=950x+481150. To determine where the line intersects the x-axis, solve 0=950x+481150. The solution is x=48127. The missile intersects the x-axis at the point (48127,0).

Find the equation of the line tangent to the hyperbola x2y2=16 at the point (5,3).

Hint

Using implicit differentiation, you should find that dydx=xy.

Solution

y=53x163

Key Concepts

  • We use implicit differentiation to find derivatives of implicitly defined functions (functions defined by equations).
  • By using implicit differentiation, we can find the equation of a tangent line to the graph of a curve.

For the following exercises, use implicit differentiation to find dydx.

1.  x2y2=4

2.  6x2+3y2=12

Solution

dydx=2xy

3.  x2y=y7

4.  3x3+9xy2=5x3

Solution

dydx=x3yy2x

5.  xycos(xy)=1

6.  yx+4=xy+8

Solution

dydx=yy2x+4x+4x

7.  xy2=x7

8.  ysin(xy)=y2+2

Solution

dydx=y2cos(xy)2ysin(xy)xycosxy

9.  (xy)2+3x=y2

10.  x3y+xy3=8

dydx=3x2yy3x3+3xy2

11. x2y+xy2=3x

12. ex2y=x+y

dydx=12xyex2yx2ex2y1

For the following exercises, find y(x) using implicit differentiation. Leave your final answer in terms of x and y only.

13. x3+y3=1.

14. x2+y2=2

y(x)=yx2yy2=(y2+x2)y3

15. xy2+3y=5

16. 4x=xy+2y

y(x)=(44yx+2)(x+2)(4y)(x+2)2

For the following exercises, find the equation of the tangent line to the graph of the given equation at the indicated point. Use a calculator or computer software to graph the function and the tangent line.

17. [T]  x4yxy3=2,(1,1)

18. [T]  x2y2+5xy=14,(2,1)

Solution

The graph has a crescent in each of the four quadrants. There is a straight line marked T(x) with slope -1/2 and y intercept 2.
y=12x+2

19. [T]  tan(xy)=y,(π4,1)

20. [T]  xy2+sin(πy)2x2=10,(2,3)

Solution

The graph has two curves, one in the first quadrant and one in the fourth quadrant. They are symmetric about the x axis. The curve in the first quadrant goes from (0.3, 5) to (1.5, 3.5) to (5, 4). There is a straight line marked T(x) with slope 1/(π + 12) and y intercept -(3π + 38)/(π + 12).
y=1π+12x3π+38π+12

21. [T]  xy+5x7=34y,(1,2)

22. [T]  xy+sin(x)=1,(π2,0)

Solution

The graph starts in the third quadrant near (-5, 0), remains near 0 until x = -4, at which point it decreases until it reaches near (0, -5). There is an asymptote at x = 0. The graph begins again near (0, 5) decreases to (1, 0) and then increases a little bit before decreasing to be near (5, 0). There is a straight line marked T(x) that coincides with y = 0.
y=0

23. [T] The graph of a folium of Descartes with equation 2x3+2y39xy=0 is given in the following graph.

A folium is graphed which has equation 2x3 + 2y3 – 9xy = 0. It crosses over itself at (0, 0).

  1. Find the equation of the tangent line at the point (2,1). Graph the tangent line along with the folium.
  2. Find the equation of the normal line to the tangent line in a. at the point (2,1).

24.  For the equation x2+2xy3y2=0,

  1. Find the equation of the normal to the tangent line at the point (1,1).
  2. At what other point does the normal line in a. intersect the graph of the equation?

Solution

a. y=x+2
b. (3,1)

25.  Find all points on the graph of y327y=x290 at which the tangent line is vertical.

26.  For the equation x2+xy+y2=7,

  1. Find the x-intercept(s).
  2. Find the slope of the tangent line(s) at the x-intercept(s).
  3. What does the value(s) in b. indicate about the tangent line(s)?

Solution

a. (±7,0)
b. Slope is -2 at both intercepts
c. They are parallel since the slope is the same at both intercepts.

27.  Find the equation of the tangent line to the graph of the equation sin1x+sin1y=π6 at the point (0,12).

28.  Find the equation of the tangent line to the graph of the equation tan1(x+y)=x2+π4 at the point (0,1).

Solution

y=x+1

29.  Find y and y for x2+6xy2y2=3.

30. [T] The number of cell phones produced when x dollars is spent on labor and y dollars is spent on capital invested by a manufacturer can be modeled by the equation 60x3/4y1/4=3240.

  1. Find dydx and evaluate at the point (81,16).
  2. Interpret the result of a.

Solution

a. -0.5926
b. When $81 is spent on labor and $16 is spent on capital, the amount spent on capital is decreasing by $0.5926 per $1 spent on labor.

31. [T] The number of cars produced when x dollars is spent on labor and y dollars is spent on capital invested by a manufacturer can be modeled by the equation 30x1/3y2/3=360.

(Both x and y are measured in thousands of dollars.)

  1. Find dydx and evaluate at the point (27,8).
  2. Interpret the result of a.

32.  The volume of a right circular cone of radius x and height y is given by V=13πx2y. Suppose that the volume of the cone is 85πcm3. Find dydx when x=4 and y=16.

Solution

dydx=8

For the following exercises, consider a closed rectangular box with a square base with side x and height y.

33.  Find an equation for the surface area of the rectangular box, S(x,y).

34.  If the surface area of the rectangular box is 78 square feet, find dydx when x=3 feet and y=5 feet.

Solution

dydx=2.67

For the following exercises, use implicit differentiation to determine y. Does the answer agree with the formulas we have previously determined?

35.  x=siny

36.  x=cosy

Solution

y=11x2

37.  x=tany

Glossary

implicit differentiation
is a technique for computing dydx for a function defined by an equation, accomplished by differentiating both sides of the equation (remembering to treat the variable y as a function) and solving for dydx

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