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Applying implicit differentiation

In a simple video game, a rocket travels in an elliptical orbit whose path is described by the equation 4 x 2 + 25 y 2 = 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 , 8 3 ) , where will it intersect the x -axis?

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

4 x 2 + 25 y 2 = 100 at ( 3 , 8 3 ) intersects the x -axis. Begin by finding d y d x implicitly.

Differentiating, we have

8 x + 50 y d y d x = 0 .

Solving for d y d x , we have

d y d x = 4 x 25 y .

The slope of the tangent line is d y d x | ( 3 , 8 3 ) = 9 50 . The equation of the tangent line is y = 9 50 x + 183 200 . To determine where the line intersects the x -axis, solve 0 = 9 50 x + 183 200 . The solution is x = 61 3 . The missile intersects the x -axis at the point ( 61 3 , 0 ) .

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Find the equation of the line tangent to the hyperbola x 2 y 2 = 16 at the point ( 5 , 3 ) .

y = 5 3 x 16 3

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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 d y d x .

6 x 2 + 3 y 2 = 12

d y d x = −2 x y

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3 x 3 + 9 x y 2 = 5 x 3

d y d x = x 3 y y 2 x

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y x + 4 = x y + 8

d y d x = y y 2 x + 4 x + 4 x

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y sin ( x y ) = y 2 + 2

d y d x = y 2 cos ( x y ) 2 y sin ( x y ) x y cos x y

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x 3 y + x y 3 = −8

d y d x = −3 x 2 y y 3 x 3 + 3 x y 2

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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.

[T] x 4 y x y 3 = −2 , ( −1 , −1 )

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[T] x 2 y 2 + 5 x y = 14 , ( 2 , 1 )


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 = −1 2 x + 2

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[T] tan ( x y ) = y , ( π 4 , 1 )

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[T] x y 2 + sin ( π y ) 2 x 2 = 10 , ( 2 , −3 )


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 = 1 π + 12 x 3 π + 38 π + 12

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[T] x y + 5 x 7 = 3 4 y , ( 1 , 2 )

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[T] x y + sin ( x ) = 1 , ( π 2 , 0 )


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

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[T] The graph of a folium of Descartes with equation 2 x 3 + 2 y 3 9 x y = 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 ) .
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For the equation x 2 + 2 x y 3 y 2 = 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?

a. y = x + 2 b. ( 3 , −1 )

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Find all points on the graph of y 3 27 y = x 2 90 at which the tangent line is vertical.

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For the equation x 2 + x y + y 2 = 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)?

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

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Find the equation of the tangent line to the graph of the equation sin −1 x + sin −1 y = π 6 at the point ( 0 , 1 2 ) .

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Find the equation of the tangent line to the graph of the equation tan −1 ( x + y ) = x 2 + π 4 at the point ( 0 , 1 ) .

y = x + 1

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Find y and y for x 2 + 6 x y 2 y 2 = 3 .

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[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 60 x 3 / 4 y 1 / 4 = 3240 .

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

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.

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[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 30 x 1 / 3 y 2 / 3 = 360 .

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

  1. Find d y d x and evaluate at the point ( 27 , 8 ) .
  2. Interpret the result of a.
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The volume of a right circular cone of radius x and height y is given by V = 1 3 π x 2 y . Suppose that the volume of the cone is 85 π cm 3 . Find d y d x when x = 4 and y = 16 .

−8

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

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

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

−2.67

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

x = cos y

y = 1 1 x 2

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Practice Key Terms 1

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Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
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